Links:

Examples of Stan code for regression models

What is in the stan fit object?

1 Data set Howell

Dataset Howell from the rethinking package.

data(Howell1)
howell1 <- Howell1
head(howell1)
##    height   weight age male
## 1 151.765 47.82561  63    1
## 2 139.700 36.48581  63    0
## 3 136.525 31.86484  65    0
## 4 156.845 53.04191  41    1
## 5 145.415 41.27687  51    0
## 6 163.830 62.99259  35    1
str(howell1)
## 'data.frame':    544 obs. of  4 variables:
##  $ height: num  152 140 137 157 145 ...
##  $ weight: num  47.8 36.5 31.9 53 41.3 ...
##  $ age   : num  63 63 65 41 51 35 32 27 19 54 ...
##  $ male  : int  1 0 0 1 0 1 0 1 0 1 ...
ggplot(howell1, aes(height),bins=50) + geom_histogram(aes(y = ..count..), fill="darkgreen",binwidth = 1, alpha=0.4) + 
  theme(axis.line = element_line(size=1, colour = "black"),
              panel.grid.major = element_line(colour = "#d3d3d3"),
              panel.grid.minor = element_blank())

Subset with only individuals over 18 years old:

howell2 <- howell1[howell1$age >= 18 , ]
plot_grid(ggplot(howell2, aes(height),bins=50) + geom_histogram(aes(y = ..count..), fill="darkgreen",binwidth = 1, alpha=0.4) + 
  theme(axis.line = element_line(size=1, colour = "black"),
              panel.grid.major = element_line(colour = "#d3d3d3"),
              panel.grid.minor = element_blank()),
  ggplot(howell2, aes(weight),bins=50) + geom_histogram(aes(y = ..count..), fill="blue",binwidth = 1, alpha=0.4) + 
  theme(axis.line = element_line(size=1, colour = "black"),
              panel.grid.major = element_line(colour = "#d3d3d3"),
              panel.grid.minor = element_blank()))

ggplot(howell2,aes(y=height,x=weight))+geom_point() + theme(axis.line = element_line(size=1, colour = "black"),
              panel.grid.major = element_line(colour = "gray95"),
              panel.grid.minor = element_blank())

2 Without predictor

2.1 With large variance

\[ h_{i} \sim \mathcal{N}(\mu \, , \, \sigma^2)\] \[\mu \sim \mathcal{N}(178,20)\]

Data in a list:

m1.data <- list(N = nrow(howell2), height = howell2$height)

2.1.1 Notation 1 (vectorized with target += notation)

explicitly incrementing the log probability function

https://mc-stan.org/docs/2_18/reference-manual/increment-log-prob-section.html

“The basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters; this function returns the log density of the posterior up to an additive constant. Data and transformed data are fixed before the log density is evaluated. The total log probability is initialized to zero. Next, any log Jacobian adjustments accrued by the variable constraints are added to the log density (the Jacobian adjustment may be skipped for optimization). Sampling and log probability increment statements may add to the log density in the model block. A log probability increment statement directly increments the log density with the value of an expression as follows.”

https://stackoverflow.com/questions/40289457/stan-using-target-syntax

“The target density is the density from which the sampler samples and it needs to be equal to the joint density of all the parameters given the data up to a constant (which is usually achieved via Bayes’s rule by coding as the joint density of parameters and modeled data up to a constant).”

data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real<lower=0,upper=50> sigma;
}
model {
  target += normal_lpdf(height | mu, sigma);       // likelihood
  target += normal_lpdf(mu | 178, 20);             // prior of the mean
}
fit.m1.1 <- sampling(m1.1, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: bb69f84792e053a2e0234193f5617c02.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##           mean se_mean   sd      10%      50%      90% n_eff Rhat
## mu      154.60    0.01 0.39   154.10   154.61   155.09   992    1
## sigma     7.77    0.01 0.30     7.39     7.76     8.14  1086    1
## lp__  -1223.07    0.04 0.92 -1224.25 -1222.81 -1222.22   523    1
## 
## Samples were drawn using NUTS(diag_e) at Thu Nov  7 13:58:01 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

2.1.2 Notation 2 (vectorized with Jags-like notation)

using the sampling statement

Difference with explicitly incrementing the log probability function:

“The sampling statement drops all the terms in the log probability function that are constant, whereas the explicit call to normal_lpdf adds all of the terms in the definition of the log normal probability function, including all of the constant normalizing terms. Therefore, the explicit increment form can be used to recreate the exact log probability values for the model. Otherwise, the sampling statement form will be faster if any of the input expressions involve only constants, data variables, and transformed data variables.”

https://mc-stan.org/docs/2_18/reference-manual/sampling-statements-section.html

Bounding sigma in the parameter block:

data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real<lower=0,upper=50> sigma;
}
model {
  height ~ normal(mu,sigma);                       // likelihood
  mu ~ normal(178, 20);                            // prior of the mean
}
fit.m1.2.1 <- sampling(m1.2.1, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.2.1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 362449507f9771e1464afdb2ddaff73e.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##          mean se_mean   sd     10%     50%     90% n_eff Rhat
## mu     154.62    0.01 0.40  154.09  154.63  155.14  1003    1
## sigma    7.76    0.01 0.29    7.40    7.76    8.13   904    1
## lp__  -895.71    0.04 0.90 -896.94 -895.48 -894.85   455    1
## 
## Samples were drawn using NUTS(diag_e) at Thu Nov  7 13:58:30 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Specifying the prior of sigma in the model block (\(\sigma \sim \mathcal{U}(0,50)\)):

data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real sigma;
}
model {
  height ~ normal(mu,sigma);                       // likelihood
  mu ~ normal(178, 20);                            // prior of the mean
  sigma ~ uniform(0,50);                           // prior of sigma          
}
fit.m1.2.2 <- sampling(m1.2.2, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.2.2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: a7cb2ed5bc279b99bc559cfdb30b62e2.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##          mean se_mean   sd     10%     50%     90% n_eff Rhat
## mu     154.60    0.02 0.41  154.11  154.60  155.13   708 1.00
## sigma    7.77    0.01 0.30    7.40    7.75    8.17   779 1.00
## lp__  -895.75    0.05 1.01 -897.15 -895.44 -894.84   447 1.01
## 
## Samples were drawn using NUTS(diag_e) at Thu Nov  7 13:59:01 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

2.1.3 Notation 3 (unvectorized)

data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real<lower=0,upper=50> sigma;
}
model {
  for (n in 1:N)
  height[n] ~ normal(mu,sigma);                    // likelihood
  mu ~ normal(178, 20);                            // prior of the mean
}
fit.m1.3 <- sampling(m1.3, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.3, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 2c0d3bdaed726863b2342ed270e93080.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##          mean se_mean   sd     10%     50%     90% n_eff Rhat
## mu     154.62    0.01 0.41  154.10  154.62  155.16  1015 1.00
## sigma    7.77    0.01 0.29    7.38    7.76    8.14  1083 1.00
## lp__  -895.74    0.04 0.94 -896.94 -895.49 -894.85   479 1.01
## 
## Samples were drawn using NUTS(diag_e) at Thu Nov  7 13:59:31 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

2.1.3.1 Comparing model runtime

Vectorized with target += notation:

get_elapsed_time(fit.m1.1)
##           warmup   sample
## chain:1 0.017447 0.007627
## chain:2 0.015749 0.006771

Vectorized with Jags-like notation:

# not specifying the prior of sigma
get_elapsed_time(fit.m1.2.1)
##           warmup   sample
## chain:1 0.015877 0.007393
## chain:2 0.017561 0.008066
# Specifying the prior of sigma
get_elapsed_time(fit.m1.2.2)
##           warmup   sample
## chain:1 0.018798 0.008674
## chain:2 0.014649 0.007995

Unvectorized:

get_elapsed_time(fit.m1.3)
##           warmup   sample
## chain:1 0.055097 0.039766
## chain:2 0.054672 0.028902

2.2 With low variance

\[ h_{i} \sim \mathcal{N}(\mu \, , \, \sigma^2)\] \[\mu \sim \mathcal{N}(178,0.1)\]

Data in a list:

m2.data <- list(N = nrow(howell2), height = howell2$height)
data {                                             // observed variables 
  int<lower=1> N;                                  // number of observations
  vector[N] height;                                // response variables
}
parameters {                                       // unobserved variables
  real mu;                                      
  real<lower=0,upper=50> sigma;
}
model {
  target += normal_lpdf(height | mu, sigma);       // likelihood
  target += normal_lpdf(mu | 178, 0.1);             // prior of the mean
}
fit.m2 <- sampling(m2 , data = m2.data, iter = 1000, chains = 2, cores = 2)
print(fit.m2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 732cdcf02bdc315da3f0b397f4c17596.
## 2 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=1000.
## 
##           mean se_mean   sd      10%      50%      90% n_eff Rhat
## mu      177.87    0.00 0.10   177.74   177.86   178.00   935    1
## sigma    24.63    0.03 0.91    23.45    24.59    25.85   909    1
## lp__  -1623.63    0.05 0.98 -1624.97 -1623.34 -1622.77   424    1
## 
## Samples were drawn using NUTS(diag_e) at Thu Nov  7 14:00:01 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3 With one continious predictor

Model equation:

\[\begin{align*} h_{i} \sim & \, \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} = & \, \beta_{1} + \beta_{2} w_{i}\\[4pt] \beta_{1} \sim & \, \mathcal{N}(178,100)\\[4pt] \beta_{2} \sim & \, \mathcal{N}(0,10)\\[4pt] \sigma \sim & \, \mathcal{U}(0,5)\\[4pt] \end{align*}\]

Centering the predictor:

howell2$weight.c <- howell2$weight - mean(howell2$weight)

Data in a list:

m3.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.c)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}
parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slopes
  real sigma;
}
model {
  height ~ normal(beta[1] + beta[2]*weight ,sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ normal(0, 10);                                 // prior of the slope
  sigma ~ uniform(0,50);
}
fit.m3.1 <- sampling(m3.1, data = m3.data, iter = 1000, chains = 4, cores = 4)
print(fit.m3.1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 2c0bf7468a6290532584633469739fd9.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##            mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]  154.60    0.01 0.28  154.25  154.60  154.95  2358    1
## beta[2]    0.91    0.00 0.04    0.85    0.90    0.96  1613    1
## sigma      5.11    0.00 0.20    4.87    5.10    5.36  2058    1
## lp__    -747.63    0.04 1.26 -749.20 -747.29 -746.40   992    1
## 
## Samples were drawn using NUTS(diag_e) at Thu Nov  7 15:43:53 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Another possibility for \(\beta_{2}\) prior: \[\beta_{2} \sim \, \text{LogNormal}(0,10)\].

data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}
parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slopes
  real sigma;
}
model {
  height ~ normal(beta[1] + beta[2]*weight ,sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ lognormal(0, 1);                               // prior of the slope
  sigma ~ uniform(0,50);
}
fit.m3.2 <- sampling(m3.2, data = m3.data, iter = 1000, chains = 4, cores = 4)
print(fit.m3.2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 33297718f34e2620ae7a419b4225e246.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##            mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]  154.61    0.01 0.27  154.25  154.61  154.95  1921    1
## beta[2]    0.90    0.00 0.04    0.85    0.90    0.96  2063    1
## sigma      5.11    0.00 0.20    4.85    5.10    5.36  2089    1
## lp__    -747.51    0.04 1.24 -749.16 -747.20 -746.28  1049    1
## 
## Samples were drawn using NUTS(diag_e) at Thu Nov  7 15:44:27 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

vectorized with target +=

data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}
parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slopes
  real sigma;
}
model {
  target += normal_lpdf(height | beta[1] + beta[2]*weight, sigma);       // likelihood
  target += normal_lpdf(beta[1] | 178, 20);                              // prior of the intercept
  target += lognormal_lpdf(beta[2] |0, 1);                                // prior of the slope
  target += uniform_lpdf(sigma|0,50);
}
fit.m3.3 <- sampling(m3.3, data = m3.data, iter = 1000, chains = 4, cores = 4)
print(fit.m3.3, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 75d942089dad056ec9f4c75cf887e097.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##             mean se_mean   sd      10%      50%      90% n_eff Rhat
## beta[1]   154.60    0.01 0.27   154.26   154.59   154.94  1705    1
## beta[2]     0.90    0.00 0.04     0.85     0.91     0.96  1460    1
## sigma       5.10    0.00 0.19     4.87     5.09     5.35  1945    1
## lp__    -1080.32    0.04 1.22 -1081.92 -1080.02 -1079.17   881    1
## 
## Samples were drawn using NUTS(diag_e) at Thu Nov  7 18:39:08 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3.1 What do our priors imply?

Let’s simulate the prior predictive distribution (simulate heights from the model, using only the priors).

See Chapter 4 of Statistical Rethinking.

set.seed(2971)
N <- 100 # 100 lines
a <- rnorm( N , 178 , 20 )
b1 <- rnorm( N , 0 , 10 )
b2 <- rlnorm( N, 0 , 1 )
dens(rnorm(  1e4, 0 , 10 ),main="b ~ dnorm(0,10)")

dens(rlnorm( 1e4 , 0 , 1 ),main="log(b) ~ dnorm(0,1)")

p <- plot( NULL , xlim=range(howell2$weight) , ylim=c(-100,400) ,xlab="weight" , ylab="height")
abline(h=0,lty=2)
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(howell2$weight)
for ( i in 1:N ) curve( a[i] + b1[i]*(x - xbar) ,
from=min(howell2$weight) , to=max(howell2$weight) , add=TRUE ,
col=col.alpha("black",0.2) )

McElreath: “For reference, I’ve added a dashed line at zero—no one is shorter than zero—and the “Wadlow” line at 272cm for the world’s tallest person. The pattern doesn’t look like any human population at all. It essentially says that the relationship between weight and height could be absurdly positive or negative. Before we’ve even seen the data, this is a bad model.”

plot( NULL , xlim=range(howell2$weight) , ylim=c(-100,400) ,xlab="weight" , ylab="height")
abline(h=0,lty=2)
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "log(b) ~ dnorm(0,1)" )
xbar <- mean(howell2$weight)
for ( i in 1:N ) curve( a[i] + b2[i]*(x - xbar) ,
from=min(howell2$weight) , to=max(howell2$weight) , add=TRUE ,
col=col.alpha("black",0.2) )

McElreath: “This is much more sensible. There is still a rare impossible relationship. But nearly all lines in the joint prior for \(\beta_{1}\) and \(\beta_{2}\) are now within human reason. We’re fussing about this prior, even though as you’ll see in the next section there is so much data in this example that the priors end up not mattering. We fuss for two reasons. First, there are many analyses in which no amount of data makes the prior irrelevant. In such cases, non-Bayesian procedures are no better off. They also depend upon structural features of the model. Paying careful attention to those features is essential. Second, thinking about the priors helps us develop better models, maybe even eventually going beyond geocentrism.”

Help to choose your prior with Stan: https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations

3.2 Convergence diagnostics

https://github.com/stan-dev/stan/wiki/Stan-Best-Practices

3.2.1 Rhat < 1.1

“The best generic diagnostic for the accuracy of a Markov chain Monte Carlo algorithm is the split R-hat statistic which quantifies the consistency of an ensemble of Markov chains. The idea is to run multiple Markov chains, 4 is an okay default but running more makes the diagnostic more sensitive, and ensure that they’re all exploring the same regions of parameter space. If even one chain is inconsistent with the others then all of the chains are suspect! In practice we have found that requiring Rhat < 1.1 is a good default requirement for each parameter.”

rhats <- rhat(fit.m3.2)
print(rhats)
##   beta[1]   beta[2]     sigma      lp__ 
## 0.9988572 0.9988693 0.9996734 1.0029385
mcmc_rhat(rhats)  + yaxis_text(hjust = 1)

3.2.2 Effective sample size

“Another potential problem to keep in mind is Markov chains that explore very slowly. Slow chains lead to low numbers of effective samples, and if there are too few effective samples then we can’t accurate estimate the number of effective samples at all. A good check for such issues is the number of effective samples per iteration – if N_eff / N < 0.001 then you should be suspect of the effective sample size calculation.”

ratios <- neff_ratio(fit.m3.2)
print(ratios)
##   beta[1]   beta[2]     sigma      lp__ 
## 0.9604525 1.0316468 1.0444858 0.5245081
mcmc_neff(ratios, size = 2)

3.2.3 MCMC diagnostics using the bayesplot package

We need:

# The posterior draws
posterior <- as.array(fit.m3.2)

# The  log of the posterior density for each draw 
lp <- log_posterior(fit.m3.2)
head(lp)
##   Iteration     Value Chain
## 1         1 -746.9374     1
## 2         2 -747.6230     1
## 3         3 -748.4853     1
## 4         4 -747.0805     1
## 5         5 -746.8393     1
## 6         6 -746.6638     1
# Some NUTS-specific diagnostic values 
np <- nuts_params(fit.m3.2)
head(np)
##   Iteration     Parameter     Value Chain
## 1         1 accept_stat__ 0.8870933     1
## 2         2 accept_stat__ 0.7779165     1
## 3         3 accept_stat__ 0.8750988     1
## 4         4 accept_stat__ 1.0000000     1
## 5         5 accept_stat__ 0.9889832     1
## 6         6 accept_stat__ 0.9230367     1

3.2.3.1 Divergent transitions

https://mc-stan.org/bayesplot/articles/visual-mcmc-diagnostics.html

mcmc_parcoord(posterior, np = np,pars=c("beta[1]","beta[2]","sigma"))

Pairs plots:

mcmc_pairs(posterior, np = np, pars = c("beta[1]","beta[2]","sigma"),  off_diag_args = list(size = 0.75))

Trace plots:

mcmc_trace(posterior, pars = "beta[2]", np = np) +  xlab("Post-warmup iteration")
## No divergences to plot.

3.3 Posterior predictive distributions

https://mc-stan.org/docs/2_18/stan-users-guide/prediction-forecasting-and-backcasting.html

m4.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.c)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}

parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slope
  real sigma;
}

model {
  height ~ normal(beta[1] + beta[2]*weight ,sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ lognormal(0, 1);                                  // prior of the slope
  sigma ~ uniform(0,50);
}

generated quantities {
  vector[N] y_rep;

  for(i in 1:N)
  y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i], sigma);
}
fit.m4 <- sampling(m4, data = m4.data, iter = 1000, chains = 4, cores = 4)
y_rep <- as.matrix(fit.m4, pars = "y_rep")
dim(y_rep)
## [1] 2000  352
ppc_dens_overlay(howell2$height, y_rep)

plot_grid(ppc_stat(y = howell2$height, yrep = y_rep, stat = "mean"),
          ppc_scatter_avg(y = howell2$height, yrep = y_rep))

3.4 Visualizing the variance

See section 4.4 of Statistical rethinking

Model with non-centered weight:

m3.1.bis.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}
parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slopes
  real sigma;
}
model {
  height ~ normal(beta[1] + beta[2]*weight ,sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ normal(0, 10);                                 // prior of the slope
  sigma ~ uniform(0,50);
}
fit.m3.1.bis <- sampling(m3.1.bis, data = m3.1.bis.data, iter = 1000, chains = 4, cores = 4)
print(fit.m3.1.bis, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: dca6bb84ed30c87a7015873cc74526bf.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##            mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]  113.90    0.08 1.96  111.32  113.86  116.45   660    1
## beta[2]    0.90    0.00 0.04    0.85    0.91    0.96   653    1
## sigma      5.11    0.01 0.20    4.86    5.11    5.37   743    1
## lp__    -749.32    0.05 1.23 -751.07 -749.03 -748.06   685    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 14:35:54 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

3.4.1 Prediction interval for average height

Figure 4.7. of Statistical rethinking

post <- as.data.frame(fit.m3.1.bis)
f_mu <- function(x) post$`beta[1]` + post$`beta[2]` * x
weight_new <- seq(25, 70)
mu <- 
  sapply(weight_new, f_mu) %>%
  tibble::as_tibble() %>%
  rename_all(function(x) weight_new) %>%
  mutate(Iter = row_number()) %>%
  gather(weight, height, -Iter) %>%
  group_by(weight) %>%
  mutate(hpdi_l = HDInterval::hdi(height, credMass = 0.89)[1],
         hpdi_h = HDInterval::hdi(height, credMass = 0.89)[2]) %>%
  mutate(mu = mean(height)) %>%
  ungroup() %>%
  mutate(weight = as.integer(weight))
p <- ggplot() 
p1 <- p +
  geom_point(data = mu %>% filter(Iter < 101),
             aes(weight, height), alpha = .05, color = 'dodgerblue') +
  labs(subtitle="Density at each weight")
p2 <- p +
  geom_point(data = howell2,
             aes(weight, height), shape = 1, color = 'dodgerblue') +
  geom_ribbon(data = mu,
              aes(x = weight, ymin = hpdi_l, ymax = hpdi_h),
              alpha = .2) +
  geom_abline(data = post,
              aes(intercept = mean(`beta[1]`), slope = mean(`beta[2]`))) +
  labs(subtitle="HPDI Interval = 0.89")
grid.arrange(p1, p2, nrow = 1)  

“Left: The first 100 values in the distribution of \(\beta_{1}\) (intercept) at each weight value. Right: The !Kung height data again, now with 89% HPDI of the mean indicated by the shaded region. Compare this region to the distributions of blue points on the left.”

3.4.2 Prediction interval for actual heights

Figure 4.8. of Statistical rethinking

Incorporating the uncertainty related to the standard deviation \(\sigma^{2}\).

sim_ht <- 
  sapply(weight_new,
         function(x)
           rnorm(NROW(post),
                 post$`beta[1]` + post$`beta[2]` * x,
                 post$sigma)) %>%
  as_tibble() %>%
  rename_all(function(x) weight_new) %>%
  mutate(Iter = row_number()) %>%
  gather(weight, height, -Iter) %>%
  group_by(weight) %>%
  mutate(pi_l = rethinking::PI(height, prob = 0.89)[1],
         pi_h = rethinking::PI(height, prob = 0.89)[2]) %>%
  ungroup() %>%
  mutate(weight = as.integer(weight))
p2 + geom_ribbon(data = sim_ht,
                 mapping = aes(x=weight, ymin=pi_l, ymax=pi_h), alpha = .1) +
  labs(subtitle = 'Prediction Intervals = 0.89')

Figure 4.8. 89% prediction interval for height, as a function of weight. The solid line is the MAP estimate of the mean height at each weight. The two shaded regions show different 89% plausible regions. The narrow shaded interval around the line is the distribution of \(\beta_{1}\) (intercept) . The wider shaded region represents the region within which the model expects to find 89% of actual heights in the population, at each weight.”"

4 With one categorical predictor

Individual sex is a categorical variable with value 0 whenever the person is a female and a value 1 whenever the person is a male.

ggplot(howell2, aes(x=as.factor(male), y=height, color=as.factor(male))) +
  geom_boxplot()   + scale_x_discrete(labels=c("0" = "Female", "1" = "Male")) + 
  theme(axis.title.x=element_blank(),legend.position = "none")

There are two ways to include binary categorical variables, either using indicator variable or index variable. More details in section 5.3.1 of Statistical rethinking (McElreath’s book).

4.1 Indicator variable (or dummy variable)

Using indicator variables is ok with binary variables, but shouldn’t be used for variables with many categories (see 5.3.2 section of Statistical rethiking).

Model equation (like in McElreath’s book):

\[\begin{align*} h_{i} &\sim \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} &= \alpha + \beta_{m} m_{i}\\[4pt] \alpha &\sim \mathcal{N}(178,20)\\[4pt] \beta_{m} &\sim \mathcal{N}(0,10)\\[4pt] \sigma &\sim \mathcal{U}(0,50)\\[4pt] \end{align*}\]

In R, we can convert our categorical variable in a dummy variable like this:

X <- model.matrix(~male, howell2)
m5.1.data <- list(N = nrow(howell2), height = howell2$height, X = X, n = length(unique(howell2$male)))
str(m5.1.data)
## List of 4
##  $ N     : int 352
##  $ height: num [1:352] 152 140 137 157 145 ...
##  $ X     : num [1:352, 1:2] 1 1 1 1 1 1 1 1 1 1 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:352] "1" "2" "3" "4" ...
##   .. ..$ : chr [1:2] "(Intercept)" "male"
##   ..- attr(*, "assign")= int [1:2] 0 1
##  $ n     : int 2
data {                                                     
  int<lower=0> N;                                          // number of observations
  int<lower=0> n;                                          // number of levels of the categorical variable
  vector[N] height;                                        // response variables
  matrix[N,n] X;                                           // categorical variable
}

parameters {                                          
  vector[n] beta;                                         // intercepts of the categorical variable
  real sigma;
}

transformed parameters {
vector[N] mu;                                             
mu = X*beta;  
}

model {
  height ~ normal(mu ,sigma);                              // likelihood
  sigma ~ uniform(0,50);
  beta[1] ~ normal(178,20);
  beta[2] ~ normal(0,10);
}

generated quantities {
  vector[N] y_rep;
 for(i in 1:N)
 y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_1_1 <- sampling(m5.1.1, data = m5.1.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_1_1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: a89cc41ee4ed1022b476f4484ab5720d.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## beta[2]      10.83    0.03 0.60   10.09   10.85   11.58   376 1.00
## sigma         5.57    0.00 0.21    5.29    5.56    5.84  1893 1.00
## mu[1]       160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[2]       149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[3]       149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[4]       160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[5]       149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[6]       160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[7]       149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[8]       160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[9]       149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[10]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[11]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[12]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[13]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[14]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[15]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[16]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[17]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[18]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[19]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[20]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[21]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[22]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[23]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[24]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[25]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[26]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[27]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[28]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[29]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[30]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[31]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[32]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[33]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[34]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[35]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[36]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[37]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[38]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[39]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[40]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[41]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[42]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[43]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[44]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[45]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[46]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[47]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[48]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[49]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[50]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[51]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[52]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[53]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[54]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[55]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[56]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[57]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[58]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[59]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[60]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[61]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[62]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[63]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[64]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[65]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[66]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[67]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[68]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[69]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[70]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[71]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[72]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[73]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[74]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[75]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[76]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[77]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[78]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[79]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[80]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[81]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[82]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[83]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[84]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[85]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[86]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[87]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[88]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[89]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[90]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[91]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[92]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[93]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[94]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[95]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[96]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[97]      149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[98]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[99]      160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[100]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[101]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[102]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[103]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[104]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[105]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[106]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[107]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[108]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[109]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[110]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[111]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[112]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[113]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[114]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[115]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[116]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[117]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[118]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[119]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[120]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[121]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[122]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[123]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[124]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[125]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[126]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[127]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[128]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[129]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[130]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[131]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[132]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[133]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[134]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[135]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[136]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[137]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[138]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[139]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[140]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[141]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[142]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[143]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[144]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[145]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[146]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[147]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[148]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[149]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[150]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[151]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[152]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[153]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[154]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[155]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[156]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[157]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[158]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[159]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[160]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[161]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[162]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[163]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[164]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[165]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[166]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[167]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[168]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[169]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[170]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[171]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[172]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[173]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[174]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[175]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[176]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[177]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[178]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[179]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[180]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[181]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[182]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[183]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[184]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[185]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[186]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[187]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[188]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[189]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[190]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[191]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[192]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[193]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[194]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[195]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[196]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[197]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[198]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[199]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[200]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[201]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[202]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[203]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[204]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[205]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[206]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[207]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[208]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[209]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[210]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[211]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[212]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[213]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[214]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[215]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[216]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[217]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[218]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[219]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[220]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[221]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[222]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[223]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[224]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[225]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[226]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[227]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[228]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[229]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[230]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[231]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[232]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[233]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[234]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[235]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[236]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[237]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[238]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[239]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[240]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[241]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[242]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[243]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[244]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[245]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[246]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[247]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[248]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[249]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[250]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[251]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[252]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[253]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[254]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[255]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[256]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[257]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[258]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[259]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[260]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[261]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[262]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[263]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[264]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[265]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[266]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[267]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[268]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[269]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[270]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[271]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[272]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[273]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[274]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[275]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[276]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[277]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[278]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[279]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[280]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[281]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[282]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[283]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[284]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[285]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[286]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[287]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[288]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[289]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[290]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[291]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[292]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[293]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[294]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[295]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[296]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[297]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[298]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[299]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[300]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[301]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[302]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[303]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[304]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[305]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[306]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[307]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[308]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[309]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[310]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[311]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[312]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[313]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[314]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[315]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[316]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[317]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[318]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[319]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[320]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[321]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[322]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[323]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[324]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[325]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[326]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[327]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[328]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[329]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[330]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[331]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[332]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[333]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[334]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[335]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[336]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[337]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[338]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[339]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[340]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[341]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[342]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[343]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[344]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[345]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[346]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[347]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[348]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[349]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[350]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## mu[351]     149.53    0.02 0.40  149.01  149.52  150.05   630 1.00
## mu[352]     160.36    0.02 0.43  159.82  160.36  160.91   564 1.00
## y_rep[1]    160.28    0.12 5.58  153.28  160.29  167.40  2102 1.00
## y_rep[2]    149.51    0.13 5.71  142.28  149.45  156.94  1966 1.00
## y_rep[3]    149.37    0.12 5.49  142.17  149.50  156.50  2013 1.00
## y_rep[4]    160.18    0.13 5.64  153.01  160.25  167.34  1954 1.00
## y_rep[5]    149.74    0.12 5.63  142.35  149.82  156.83  2138 1.00
## y_rep[6]    160.22    0.12 5.67  152.95  160.31  167.10  2180 1.00
## y_rep[7]    149.36    0.13 5.60  142.37  149.22  156.34  1753 1.00
## y_rep[8]    160.47    0.14 5.69  153.16  160.45  167.95  1703 1.00
## y_rep[9]    149.49    0.13 5.65  142.24  149.48  156.83  1862 1.00
## y_rep[10]   160.49    0.14 5.58  153.52  160.47  167.57  1682 1.00
## y_rep[11]   149.35    0.12 5.48  142.34  149.41  156.21  1966 1.00
## y_rep[12]   160.38    0.12 5.58  153.23  160.34  167.56  2008 1.00
## y_rep[13]   149.45    0.12 5.48  142.62  149.47  156.44  1923 1.00
## y_rep[14]   149.51    0.14 5.66  142.23  149.41  156.79  1729 1.00
## y_rep[15]   149.54    0.13 5.53  142.50  149.54  156.81  1830 1.00
## y_rep[16]   160.30    0.13 5.58  153.32  160.33  167.44  1807 1.00
## y_rep[17]   160.34    0.13 5.58  153.16  160.24  167.54  1957 1.00
## y_rep[18]   149.52    0.12 5.50  142.30  149.87  156.50  2049 1.00
## y_rep[19]   160.32    0.13 5.41  153.43  160.30  167.34  1723 1.00
## y_rep[20]   149.80    0.13 5.56  142.71  149.89  156.59  1938 1.00
## y_rep[21]   160.53    0.12 5.58  153.33  160.53  167.41  2159 1.00
## y_rep[22]   149.52    0.12 5.45  142.49  149.61  156.60  1958 1.00
## y_rep[23]   149.69    0.13 5.67  142.58  149.61  157.08  1892 1.00
## y_rep[24]   160.60    0.13 5.57  153.52  160.61  168.04  1846 1.00
## y_rep[25]   160.45    0.12 5.39  153.50  160.56  167.20  2030 1.00
## y_rep[26]   149.46    0.13 5.63  142.11  149.35  156.60  1943 1.00
## y_rep[27]   149.67    0.12 5.59  142.58  149.53  156.70  2024 1.00
## y_rep[28]   149.77    0.12 5.59  142.56  149.95  156.85  2143 1.00
## y_rep[29]   149.68    0.13 5.54  142.60  149.87  156.77  1884 1.00
## y_rep[30]   149.57    0.12 5.48  142.42  149.65  156.78  2042 1.00
## y_rep[31]   149.77    0.12 5.61  142.72  149.82  156.89  2125 1.00
## y_rep[32]   160.31    0.13 5.73  153.04  160.30  167.55  1983 1.00
## y_rep[33]   160.60    0.13 5.64  153.54  160.60  167.84  1937 1.00
## y_rep[34]   160.10    0.12 5.56  153.12  160.09  167.16  2114 1.00
## y_rep[35]   160.28    0.13 5.53  153.40  160.07  167.57  1904 1.00
## y_rep[36]   149.63    0.13 5.51  142.81  149.55  156.76  1860 1.00
## y_rep[37]   149.59    0.12 5.67  142.25  149.42  156.84  2082 1.00
## y_rep[38]   160.49    0.13 5.66  153.22  160.44  167.89  1962 1.00
## y_rep[39]   149.50    0.14 5.60  142.51  149.58  156.61  1685 1.00
## y_rep[40]   160.58    0.12 5.53  153.49  160.66  167.71  2018 1.00
## y_rep[41]   160.37    0.13 5.55  153.25  160.41  167.46  1926 1.00
## y_rep[42]   160.48    0.12 5.38  153.42  160.40  167.47  1916 1.00
## y_rep[43]   149.66    0.13 5.55  142.51  149.63  156.75  1833 1.00
## y_rep[44]   149.57    0.13 5.68  142.20  149.54  156.77  1817 1.00
## y_rep[45]   149.33    0.12 5.38  142.52  149.45  155.96  1864 1.00
## y_rep[46]   149.64    0.13 5.66  142.35  149.61  156.93  1830 1.00
## y_rep[47]   160.50    0.12 5.49  153.72  160.44  167.59  1938 1.00
## y_rep[48]   149.32    0.13 5.69  141.86  149.55  156.35  1958 1.00
## y_rep[49]   160.26    0.12 5.55  153.18  160.23  167.47  1985 1.00
## y_rep[50]   149.32    0.13 5.74  141.80  149.46  156.50  1956 1.00
## y_rep[51]   149.71    0.13 5.69  142.09  149.75  157.09  1797 1.00
## y_rep[52]   149.59    0.13 5.65  142.36  149.60  156.97  2029 1.00
## y_rep[53]   160.39    0.13 5.71  153.34  160.22  167.94  1829 1.00
## y_rep[54]   160.17    0.12 5.41  153.07  160.16  167.10  2165 1.00
## y_rep[55]   149.54    0.13 5.51  142.51  149.68  156.39  1906 1.00
## y_rep[56]   160.35    0.13 5.70  152.96  160.27  167.74  1840 1.00
## y_rep[57]   149.41    0.13 5.63  142.26  149.36  156.66  1891 1.00
## y_rep[58]   160.37    0.12 5.54  153.09  160.27  167.63  1981 1.00
## y_rep[59]   149.49    0.13 5.68  142.22  149.64  156.76  1955 1.00
## y_rep[60]   160.17    0.13 5.56  153.09  160.22  167.28  1903 1.00
## y_rep[61]   149.42    0.13 5.72  142.15  149.44  156.61  1933 1.00
## y_rep[62]   149.73    0.12 5.59  142.63  149.68  156.85  2122 1.00
## y_rep[63]   160.33    0.12 5.62  153.02  160.36  167.50  2094 1.00
## y_rep[64]   160.20    0.13 5.59  153.24  160.23  167.36  1721 1.00
## y_rep[65]   149.51    0.13 5.73  142.16  149.47  156.73  2035 1.00
## y_rep[66]   160.32    0.12 5.72  152.97  160.42  167.45  2127 1.00
## y_rep[67]   149.70    0.12 5.66  142.54  149.74  156.84  2057 1.00
## y_rep[68]   149.59    0.12 5.58  142.56  149.52  156.80  2084 1.00
## y_rep[69]   160.47    0.12 5.51  153.11  160.52  167.41  2087 1.00
## y_rep[70]   149.48    0.13 5.62  142.46  149.45  156.36  1779 1.00
## y_rep[71]   160.34    0.13 5.77  153.13  160.17  167.84  1979 1.00
## y_rep[72]   160.31    0.12 5.41  153.38  160.30  167.35  1894 1.00
## y_rep[73]   160.30    0.13 5.60  153.15  160.24  167.44  1877 1.00
## y_rep[74]   149.63    0.12 5.59  142.48  149.60  156.74  2086 1.00
## y_rep[75]   149.69    0.13 5.53  142.52  149.60  156.67  1751 1.00
## y_rep[76]   149.51    0.13 5.54  142.42  149.34  156.72  1777 1.00
## y_rep[77]   149.44    0.13 5.52  142.21  149.41  156.48  1836 1.00
## y_rep[78]   160.50    0.13 5.54  153.40  160.60  167.59  1895 1.00
## y_rep[79]   149.62    0.13 5.54  142.65  149.56  156.80  1842 1.00
## y_rep[80]   149.48    0.12 5.40  142.44  149.41  156.33  1902 1.00
## y_rep[81]   160.20    0.13 5.73  152.98  160.22  167.57  2032 1.00
## y_rep[82]   160.35    0.13 5.62  153.05  160.23  167.29  1917 1.00
## y_rep[83]   149.48    0.13 5.57  142.45  149.45  156.72  1823 1.00
## y_rep[84]   160.44    0.13 5.33  153.71  160.37  167.38  1680 1.00
## y_rep[85]   149.53    0.13 5.55  142.53  149.46  156.59  1943 1.00
## y_rep[86]   160.29    0.12 5.43  153.49  160.18  167.32  2228 1.00
## y_rep[87]   149.49    0.13 5.67  142.11  149.65  156.89  1861 1.00
## y_rep[88]   160.48    0.13 5.56  153.30  160.37  167.72  1710 1.00
## y_rep[89]   149.50    0.13 5.74  142.15  149.49  156.73  2027 1.00
## y_rep[90]   160.41    0.13 5.61  153.21  160.25  167.51  1996 1.00
## y_rep[91]   160.51    0.12 5.45  153.35  160.56  167.56  1966 1.00
## y_rep[92]   160.14    0.13 5.74  152.76  160.05  167.47  2019 1.00
## y_rep[93]   149.67    0.12 5.60  142.07  149.80  156.93  2110 1.00
## y_rep[94]   149.52    0.13 5.63  142.23  149.52  156.58  1921 1.00
## y_rep[95]   160.02    0.12 5.74  152.67  160.03  167.10  2167 1.00
## y_rep[96]   160.14    0.13 5.66  152.93  160.04  167.31  2043 1.00
## y_rep[97]   149.48    0.12 5.37  142.60  149.34  156.34  1972 1.00
## y_rep[98]   160.61    0.13 5.52  153.62  160.64  167.74  1815 1.00
## y_rep[99]   160.44    0.12 5.49  153.35  160.45  167.39  2061 1.00
## y_rep[100]  149.80    0.14 5.55  142.94  149.65  157.20  1660 1.00
## y_rep[101]  160.28    0.13 5.66  153.05  160.26  167.56  2036 1.00
## y_rep[102]  149.60    0.14 5.69  142.36  149.51  157.03  1756 1.00
## y_rep[103]  149.36    0.12 5.68  141.94  149.50  156.57  2074 1.00
## y_rep[104]  160.41    0.13 5.56  153.21  160.24  167.65  1804 1.00
## y_rep[105]  160.34    0.12 5.67  153.02  160.38  167.49  2072 1.00
## y_rep[106]  149.66    0.14 5.77  142.41  149.69  157.17  1705 1.00
## y_rep[107]  149.47    0.13 5.47  142.47  149.45  156.46  1831 1.00
## y_rep[108]  149.29    0.13 5.59  142.11  149.36  156.29  1816 1.00
## y_rep[109]  160.46    0.13 5.70  153.04  160.59  167.84  1951 1.00
## y_rep[110]  149.64    0.13 5.72  142.31  149.54  156.94  1970 1.00
## y_rep[111]  160.45    0.13 5.46  153.61  160.44  167.45  1710 1.00
## y_rep[112]  149.56    0.13 5.71  142.09  149.34  157.17  1889 1.00
## y_rep[113]  149.66    0.13 5.71  142.47  149.81  156.67  2024 1.00
## y_rep[114]  160.42    0.13 5.73  153.36  160.31  167.83  1898 1.00
## y_rep[115]  160.43    0.13 5.50  153.76  160.34  167.53  1678 1.00
## y_rep[116]  149.45    0.13 5.55  142.29  149.50  156.52  1960 1.00
## y_rep[117]  160.27    0.12 5.44  153.30  160.17  167.23  1977 1.00
## y_rep[118]  160.34    0.12 5.61  153.18  160.26  167.57  2080 1.00
## y_rep[119]  149.77    0.13 5.57  142.65  149.69  156.98  1858 1.00
## y_rep[120]  149.57    0.12 5.60  142.46  149.46  156.75  2048 1.00
## y_rep[121]  160.24    0.13 5.83  152.81  160.16  167.53  1868 1.00
## y_rep[122]  149.55    0.13 5.66  142.39  149.58  156.48  2049 1.00
## y_rep[123]  160.16    0.13 5.64  152.80  160.23  167.39  1887 1.00
## y_rep[124]  149.64    0.12 5.60  142.60  149.58  156.73  2087 1.00
## y_rep[125]  149.54    0.12 5.58  142.37  149.66  156.71  2018 1.00
## y_rep[126]  149.65    0.13 5.51  142.46  149.66  156.42  1719 1.00
## y_rep[127]  149.69    0.13 5.61  142.59  149.77  156.50  1879 1.00
## y_rep[128]  160.36    0.13 5.54  153.24  160.30  167.64  1930 1.00
## y_rep[129]  149.60    0.12 5.53  142.50  149.65  156.86  1970 1.00
## y_rep[130]  149.56    0.14 5.49  142.53  149.60  156.55  1646 1.00
## y_rep[131]  160.21    0.13 5.59  152.87  160.29  167.29  1880 1.00
## y_rep[132]  160.37    0.14 5.66  153.09  160.48  167.66  1726 1.00
## y_rep[133]  149.32    0.13 5.57  142.29  149.29  156.63  1843 1.00
## y_rep[134]  160.46    0.12 5.63  153.12  160.52  167.76  2033 1.00
## y_rep[135]  149.49    0.13 5.63  142.33  149.50  156.57  1984 1.00
## y_rep[136]  160.51    0.12 5.69  153.18  160.62  167.75  2216 1.00
## y_rep[137]  160.58    0.14 5.63  153.35  160.63  167.90  1695 1.00
## y_rep[138]  160.27    0.13 5.55  153.16  160.20  167.30  1930 1.00
## y_rep[139]  149.63    0.12 5.47  142.67  149.67  156.71  2040 1.00
## y_rep[140]  160.52    0.12 5.41  153.67  160.49  167.49  1928 1.00
## y_rep[141]  149.41    0.13 5.66  142.49  149.37  156.64  1946 1.00
## y_rep[142]  149.43    0.12 5.47  142.24  149.58  156.42  1934 1.00
## y_rep[143]  149.50    0.12 5.50  142.60  149.42  156.71  1988 1.00
## y_rep[144]  149.55    0.13 5.72  142.21  149.63  156.85  1969 1.00
## y_rep[145]  149.53    0.12 5.55  142.55  149.43  156.92  2303 1.00
## y_rep[146]  160.50    0.13 5.67  153.20  160.58  167.66  1940 1.00
## y_rep[147]  149.68    0.12 5.45  142.66  149.52  157.05  1950 1.00
## y_rep[148]  149.56    0.13 5.72  142.39  149.41  157.01  2042 1.00
## y_rep[149]  160.24    0.13 5.51  153.26  160.34  167.17  1698 1.00
## y_rep[150]  149.39    0.13 5.71  142.22  149.29  156.58  1976 1.00
## y_rep[151]  160.46    0.13 5.59  153.36  160.58  167.50  1974 1.00
## y_rep[152]  149.61    0.13 5.58  142.26  149.68  156.76  1978 1.00
## y_rep[153]  160.21    0.13 5.63  152.95  160.35  167.43  1873 1.00
## y_rep[154]  149.62    0.12 5.57  142.38  149.68  156.64  1989 1.00
## y_rep[155]  160.28    0.12 5.45  153.35  160.40  167.45  2055 1.00
## y_rep[156]  149.45    0.12 5.57  142.39  149.53  156.51  2016 1.00
## y_rep[157]  149.47    0.12 5.50  142.49  149.49  156.55  2139 1.00
## y_rep[158]  160.43    0.13 5.57  153.32  160.48  167.40  1901 1.00
## y_rep[159]  160.43    0.12 5.67  153.14  160.55  167.35  2095 1.00
## y_rep[160]  149.51    0.13 5.68  142.26  149.40  156.75  1842 1.00
## y_rep[161]  160.45    0.13 5.60  153.35  160.36  167.69  1929 1.00
## y_rep[162]  149.31    0.12 5.48  142.24  149.41  156.27  2108 1.00
## y_rep[163]  160.30    0.12 5.50  153.21  160.12  167.49  2048 1.00
## y_rep[164]  160.35    0.13 5.54  153.24  160.15  167.42  1811 1.00
## y_rep[165]  160.32    0.13 5.57  153.09  160.45  167.55  1890 1.00
## y_rep[166]  149.36    0.13 5.50  142.28  149.40  156.54  1801 1.00
## y_rep[167]  160.34    0.13 5.54  153.48  160.26  167.48  1929 1.00
## y_rep[168]  149.72    0.13 5.57  142.52  149.87  156.57  1929 1.00
## y_rep[169]  149.36    0.13 5.58  141.93  149.48  156.48  1890 1.00
## y_rep[170]  149.55    0.12 5.40  142.57  149.46  156.41  2029 1.00
## y_rep[171]  149.51    0.12 5.39  142.66  149.33  156.52  1964 1.00
## y_rep[172]  160.28    0.13 5.58  153.06  160.27  167.36  1824 1.00
## y_rep[173]  149.64    0.13 5.64  142.51  149.54  156.62  1976 1.00
## y_rep[174]  160.43    0.13 5.69  153.26  160.47  167.92  1912 1.00
## y_rep[175]  149.33    0.12 5.49  142.04  149.34  156.40  2021 1.00
## y_rep[176]  160.26    0.12 5.41  153.26  160.34  167.09  1924 1.00
## y_rep[177]  149.59    0.13 5.57  142.68  149.52  156.66  1729 1.00
## y_rep[178]  149.64    0.13 5.54  142.46  149.56  156.66  1865 1.00
## y_rep[179]  160.38    0.12 5.51  153.22  160.34  167.35  2049 1.00
## y_rep[180]  149.54    0.12 5.63  142.37  149.53  156.71  2122 1.00
## y_rep[181]  160.05    0.12 5.51  152.93  160.15  167.06  1988 1.00
## y_rep[182]  149.39    0.12 5.60  142.12  149.55  156.54  2011 1.00
## y_rep[183]  149.49    0.13 5.58  142.44  149.51  156.82  1918 1.00
## y_rep[184]  149.44    0.13 5.55  142.50  149.44  156.51  1883 1.00
## y_rep[185]  160.34    0.12 5.55  153.20  160.27  167.28  2027 1.00
## y_rep[186]  160.68    0.13 5.68  153.55  160.72  168.02  1975 1.00
## y_rep[187]  149.32    0.12 5.53  142.30  149.35  156.41  2096 1.00
## y_rep[188]  160.41    0.12 5.46  153.20  160.62  167.16  2161 1.00
## y_rep[189]  149.56    0.13 5.61  142.44  149.41  156.91  1957 1.00
## y_rep[190]  160.31    0.13 5.59  153.17  160.44  167.22  1995 1.00
## y_rep[191]  149.33    0.13 5.66  142.05  149.41  156.53  1801 1.00
## y_rep[192]  160.30    0.12 5.49  153.20  160.29  167.31  1936 1.00
## y_rep[193]  149.46    0.12 5.56  142.57  149.44  156.55  2009 1.00
## y_rep[194]  160.37    0.12 5.58  153.07  160.49  167.52  2092 1.00
## y_rep[195]  149.64    0.13 5.62  142.69  149.47  156.77  1945 1.00
## y_rep[196]  149.62    0.12 5.52  142.55  149.62  156.75  1999 1.00
## y_rep[197]  149.41    0.13 5.52  142.17  149.48  156.27  1747 1.00
## y_rep[198]  160.37    0.13 5.58  153.24  160.31  167.84  1955 1.00
## y_rep[199]  160.34    0.14 5.50  153.35  160.45  167.32  1554 1.00
## y_rep[200]  149.73    0.13 5.72  142.88  149.66  157.15  1967 1.00
## y_rep[201]  160.63    0.12 5.64  153.37  160.69  167.74  2059 1.00
## y_rep[202]  149.26    0.12 5.53  142.25  149.06  156.55  1976 1.00
## y_rep[203]  160.38    0.12 5.61  153.21  160.33  167.45  2046 1.00
## y_rep[204]  149.49    0.12 5.50  142.51  149.60  156.28  2011 1.00
## y_rep[205]  160.29    0.12 5.43  153.34  160.28  167.11  1897 1.00
## y_rep[206]  149.49    0.13 5.63  142.15  149.52  156.64  1899 1.00
## y_rep[207]  149.57    0.13 5.54  142.31  149.71  156.63  1944 1.00
## y_rep[208]  149.59    0.14 5.63  142.44  149.64  156.68  1731 1.00
## y_rep[209]  149.35    0.12 5.51  142.09  149.34  156.35  2127 1.00
## y_rep[210]  149.37    0.13 5.54  142.13  149.44  156.40  1731 1.00
## y_rep[211]  160.27    0.13 5.48  153.25  160.26  167.24  1914 1.00
## y_rep[212]  160.31    0.12 5.47  153.39  160.35  167.26  1935 1.00
## y_rep[213]  160.25    0.13 5.67  153.00  160.28  167.48  2011 1.00
## y_rep[214]  149.50    0.12 5.53  142.38  149.38  156.67  2100 1.00
## y_rep[215]  160.12    0.12 5.56  152.67  160.26  167.06  2014 1.00
## y_rep[216]  160.37    0.13 5.57  153.23  160.45  167.54  1881 1.00
## y_rep[217]  160.35    0.14 5.63  153.07  160.30  167.51  1550 1.00
## y_rep[218]  149.38    0.13 5.61  142.15  149.44  156.52  1981 1.00
## y_rep[219]  160.61    0.13 5.54  153.55  160.52  167.71  1836 1.00
## y_rep[220]  149.47    0.13 5.63  142.08  149.53  156.49  1929 1.00
## y_rep[221]  149.44    0.13 5.72  142.11  149.48  156.68  2020 1.00
## y_rep[222]  149.50    0.12 5.47  142.47  149.33  156.69  1979 1.00
## y_rep[223]  160.27    0.13 5.64  153.15  160.22  167.44  2012 1.00
## y_rep[224]  149.57    0.13 5.67  142.54  149.55  156.93  2032 1.00
## y_rep[225]  160.32    0.14 5.64  153.12  160.26  167.79  1728 1.00
## y_rep[226]  149.53    0.12 5.59  142.33  149.59  156.61  2058 1.00
## y_rep[227]  149.46    0.13 5.64  142.02  149.46  156.80  2012 1.00
## y_rep[228]  160.53    0.12 5.68  153.40  160.28  168.06  2105 1.00
## y_rep[229]  149.70    0.13 5.46  142.84  149.62  156.88  1895 1.00
## y_rep[230]  160.35    0.13 5.60  153.23  160.32  167.45  1975 1.00
## y_rep[231]  160.50    0.12 5.44  153.53  160.43  167.60  1930 1.00
## y_rep[232]  149.69    0.13 5.70  142.33  149.73  157.02  1953 1.00
## y_rep[233]  149.40    0.13 5.47  142.41  149.39  156.53  1856 1.00
## y_rep[234]  160.39    0.12 5.60  153.14  160.38  167.34  2042 1.00
## y_rep[235]  160.52    0.12 5.54  153.83  160.22  167.78  2190 1.00
## y_rep[236]  149.38    0.13 5.63  141.93  149.43  156.72  1786 1.00
## y_rep[237]  160.39    0.12 5.46  153.32  160.43  167.40  2009 1.00
## y_rep[238]  149.84    0.12 5.43  142.82  149.70  156.82  2161 1.00
## y_rep[239]  149.62    0.12 5.56  142.56  149.59  156.98  2014 1.00
## y_rep[240]  149.46    0.14 5.57  142.12  149.58  156.35  1532 1.00
## y_rep[241]  149.61    0.13 5.68  142.30  149.65  157.13  1929 1.00
## y_rep[242]  160.24    0.13 5.67  153.00  160.36  167.45  1932 1.00
## y_rep[243]  160.41    0.12 5.66  153.28  160.37  167.69  2084 1.00
## y_rep[244]  149.48    0.12 5.56  142.25  149.37  156.51  1996 1.00
## y_rep[245]  160.44    0.13 5.67  153.27  160.50  167.80  1763 1.00
## y_rep[246]  149.45    0.13 5.50  142.61  149.42  156.49  1748 1.00
## y_rep[247]  160.44    0.13 5.67  153.30  160.44  167.68  1818 1.00
## y_rep[248]  149.55    0.13 5.55  142.60  149.43  156.65  1961 1.00
## y_rep[249]  160.54    0.13 5.55  153.44  160.50  167.62  1846 1.00
## y_rep[250]  160.30    0.12 5.63  153.15  160.27  167.74  2212 1.00
## y_rep[251]  160.28    0.13 5.55  153.21  160.37  167.15  1856 1.00
## y_rep[252]  160.29    0.13 5.70  152.94  160.36  167.72  1932 1.00
## y_rep[253]  149.54    0.14 5.70  142.34  149.43  156.73  1678 1.00
## y_rep[254]  149.66    0.12 5.64  142.36  149.77  156.70  2082 1.00
## y_rep[255]  149.63    0.13 5.65  142.27  149.67  156.88  1978 1.00
## y_rep[256]  149.44    0.13 5.60  142.40  149.42  156.62  1936 1.00
## y_rep[257]  160.44    0.13 5.56  153.35  160.39  167.44  1965 1.00
## y_rep[258]  149.59    0.12 5.55  142.71  149.48  156.53  2135 1.00
## y_rep[259]  160.41    0.13 5.47  153.40  160.48  167.48  1775 1.00
## y_rep[260]  160.25    0.13 5.65  153.05  160.15  167.56  1857 1.00
## y_rep[261]  160.22    0.12 5.48  153.43  160.16  167.19  1943 1.00
## y_rep[262]  149.71    0.12 5.67  142.44  149.69  156.91  2074 1.00
## y_rep[263]  149.51    0.13 5.56  142.59  149.49  156.56  1780 1.00
## y_rep[264]  149.62    0.13 5.77  142.30  149.71  156.87  2042 1.00
## y_rep[265]  160.49    0.13 5.60  153.33  160.39  167.69  1945 1.00
## y_rep[266]  149.54    0.13 5.48  142.59  149.54  156.57  1768 1.00
## y_rep[267]  160.44    0.13 5.59  153.38  160.40  167.51  1955 1.00
## y_rep[268]  149.42    0.13 5.60  142.33  149.28  156.48  1935 1.00
## y_rep[269]  149.64    0.13 5.59  142.49  149.73  156.68  1958 1.00
## y_rep[270]  160.39    0.13 5.79  152.90  160.36  167.90  1900 1.00
## y_rep[271]  160.61    0.12 5.68  153.29  160.71  167.86  2077 1.00
## y_rep[272]  149.25    0.12 5.50  142.13  149.43  156.21  1934 1.00
## y_rep[273]  160.44    0.13 5.54  153.46  160.42  167.51  1928 1.00
## y_rep[274]  160.38    0.12 5.35  153.50  160.26  167.18  1985 1.00
## y_rep[275]  149.51    0.13 5.54  142.61  149.33  156.61  1837 1.00
## y_rep[276]  160.31    0.12 5.63  153.14  160.46  167.68  2105 1.00
## y_rep[277]  149.65    0.12 5.32  142.70  149.51  156.68  1978 1.00
## y_rep[278]  149.27    0.13 5.62  141.89  149.37  156.25  1980 1.00
## y_rep[279]  160.35    0.13 5.58  153.18  160.31  167.39  1913 1.00
## y_rep[280]  149.50    0.12 5.51  142.42  149.40  156.65  1994 1.00
## y_rep[281]  149.74    0.13 5.70  142.65  149.57  157.07  1924 1.00
## y_rep[282]  160.18    0.12 5.56  153.09  160.23  167.10  2043 1.00
## y_rep[283]  149.42    0.12 5.41  142.43  149.36  156.40  2059 1.00
## y_rep[284]  160.28    0.13 5.50  153.04  160.27  167.44  1908 1.00
## y_rep[285]  149.41    0.12 5.54  142.38  149.42  156.39  2139 1.00
## y_rep[286]  160.32    0.12 5.54  153.17  160.38  167.22  2016 1.00
## y_rep[287]  160.53    0.13 5.58  153.31  160.60  167.73  1937 1.00
## y_rep[288]  160.40    0.12 5.65  153.00  160.71  167.66  2159 1.00
## y_rep[289]  149.58    0.13 5.66  142.40  149.56  156.86  2035 1.00
## y_rep[290]  160.37    0.13 5.63  152.99  160.38  167.36  1925 1.00
## y_rep[291]  160.29    0.12 5.51  153.39  160.02  167.33  1998 1.00
## y_rep[292]  160.21    0.13 5.54  153.07  160.22  167.27  1808 1.00
## y_rep[293]  160.38    0.12 5.41  153.43  160.47  167.07  2015 1.00
## y_rep[294]  149.62    0.12 5.45  142.65  149.63  156.70  2122 1.00
## y_rep[295]  160.25    0.13 5.62  153.02  160.31  167.14  2016 1.00
## y_rep[296]  149.61    0.13 5.64  142.29  149.65  156.71  1987 1.00
## y_rep[297]  149.60    0.13 5.55  142.47  149.70  156.85  1736 1.00
## y_rep[298]  149.53    0.12 5.50  142.52  149.57  156.72  2110 1.00
## y_rep[299]  160.47    0.12 5.49  153.36  160.34  167.58  2077 1.00
## y_rep[300]  160.25    0.13 5.52  153.06  160.35  167.44  1738 1.00
## y_rep[301]  149.42    0.13 5.50  142.63  149.33  156.52  1866 1.00
## y_rep[302]  160.53    0.12 5.70  153.06  160.65  167.63  2208 1.00
## y_rep[303]  149.64    0.12 5.57  142.35  149.72  156.83  2070 1.00
## y_rep[304]  160.54    0.12 5.46  153.63  160.35  167.64  2003 1.00
## y_rep[305]  160.23    0.12 5.61  152.92  160.26  167.51  2065 1.00
## y_rep[306]  160.34    0.12 5.50  153.26  160.45  167.14  2165 1.00
## y_rep[307]  160.48    0.12 5.66  153.20  160.35  167.90  2114 1.00
## y_rep[308]  149.48    0.13 5.65  142.28  149.54  156.84  2029 1.00
## y_rep[309]  149.61    0.13 5.54  142.29  149.59  156.70  1849 1.00
## y_rep[310]  149.54    0.12 5.48  142.48  149.54  156.54  2041 1.00
## y_rep[311]  149.67    0.12 5.71  142.32  149.77  156.93  2134 1.00
## y_rep[312]  149.52    0.13 5.41  142.64  149.40  156.38  1726 1.00
## y_rep[313]  160.33    0.13 5.64  153.13  160.17  167.77  1948 1.00
## y_rep[314]  149.36    0.13 5.46  142.53  149.30  156.38  1885 1.00
## y_rep[315]  160.07    0.12 5.53  152.91  160.09  167.38  2002 1.00
## y_rep[316]  149.48    0.13 5.65  142.31  149.45  156.77  1925 1.00
## y_rep[317]  160.40    0.12 5.59  153.21  160.37  167.46  2003 1.00
## y_rep[318]  149.38    0.13 5.56  142.47  149.23  156.37  1912 1.00
## y_rep[319]  160.33    0.13 5.68  153.23  160.20  167.69  1971 1.00
## y_rep[320]  149.50    0.12 5.52  142.47  149.43  156.71  2005 1.00
## y_rep[321]  160.47    0.12 5.65  153.20  160.54  167.71  2102 1.00
## y_rep[322]  149.40    0.12 5.50  142.52  149.32  156.41  2026 1.00
## y_rep[323]  160.50    0.13 5.49  153.42  160.50  167.55  1925 1.00
## y_rep[324]  160.61    0.13 5.62  153.26  160.55  167.99  1884 1.00
## y_rep[325]  149.57    0.13 5.72  142.30  149.50  156.83  2018 1.00
## y_rep[326]  149.46    0.12 5.67  142.39  149.45  156.78  2207 1.00
## y_rep[327]  149.41    0.13 5.61  142.19  149.50  156.42  1912 1.00
## y_rep[328]  160.34    0.13 5.73  152.92  160.24  167.55  1829 1.00
## y_rep[329]  160.45    0.13 5.57  153.58  160.21  167.69  1821 1.00
## y_rep[330]  160.66    0.13 5.67  153.28  160.62  167.78  1887 1.00
## y_rep[331]  160.33    0.12 5.72  152.86  160.33  167.53  2119 1.00
## y_rep[332]  160.66    0.12 5.49  153.64  160.54  167.72  2024 1.00
## y_rep[333]  149.51    0.13 5.67  142.50  149.65  156.75  1852 1.00
## y_rep[334]  160.45    0.13 5.58  153.32  160.46  167.98  1901 1.00
## y_rep[335]  149.57    0.13 5.60  142.47  149.58  156.91  1957 1.00
## y_rep[336]  160.35    0.12 5.63  153.16  160.44  167.40  2029 1.00
## y_rep[337]  149.53    0.13 5.70  142.30  149.49  156.75  2008 1.00
## y_rep[338]  149.52    0.13 5.69  142.37  149.50  157.05  2005 1.00
## y_rep[339]  149.58    0.13 5.58  142.44  149.67  156.59  1994 1.00
## y_rep[340]  149.60    0.12 5.46  142.47  149.72  156.79  2034 1.00
## y_rep[341]  149.38    0.13 5.69  142.09  149.44  156.62  1998 1.00
## y_rep[342]  160.36    0.12 5.51  153.43  160.20  167.39  2073 1.00
## y_rep[343]  149.52    0.12 5.33  142.77  149.49  156.35  2045 1.00
## y_rep[344]  160.30    0.14 5.59  152.97  160.34  167.31  1572 1.00
## y_rep[345]  149.71    0.12 5.70  142.40  149.67  157.00  2088 1.00
## y_rep[346]  160.38    0.13 5.53  153.30  160.38  167.34  1849 1.00
## y_rep[347]  149.54    0.13 5.73  141.98  149.65  156.87  2018 1.00
## y_rep[348]  149.35    0.12 5.43  142.43  149.34  156.23  1929 1.00
## y_rep[349]  149.40    0.12 5.50  142.42  149.44  156.39  1983 1.00
## y_rep[350]  160.29    0.12 5.59  153.18  160.31  167.50  2061 1.00
## y_rep[351]  149.53    0.14 5.55  142.44  149.48  156.40  1688 1.00
## y_rep[352]  160.32    0.13 5.54  153.22  160.23  167.28  1943 1.00
## lp__       -780.54    0.06 1.23 -782.17 -780.23 -779.32   423 1.02
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 15:25:38 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
summary(lm(height~as.factor(male),data=howell2))
## 
## Call:
## lm(formula = height ~ as.factor(male), data = howell2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.6585  -3.4635   0.2965   3.4715  18.7115 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      149.5135     0.4049  369.25   <2e-16 ***
## as.factor(male)1  10.8450     0.5914   18.34   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.537 on 350 degrees of freedom
## Multiple R-squared:   0.49,  Adjusted R-squared:  0.4885 
## F-statistic: 336.3 on 1 and 350 DF,  p-value: < 2.2e-16

Convergence diagnostics

posterior <- as.array(fit.m5_1_1) # posterior draws
lp <- log_posterior(fit.m5_1_1) # log of the posterior density for each draw 
np <- nuts_params(fit.m5_1_1) # Some NUTS-specific diagnostic values 
mcmc_parcoord(posterior, np = np,pars=c("beta[1]","beta[2]","sigma")) 

mcmc_pairs(posterior, np = np, pars = c("beta[1]","beta[2]","sigma"),  off_diag_args = list(size = 0.75)) # Pairs plots

mcmc_trace(posterior, pars = c("beta[1]","beta[2]","sigma"), np = np) +  xlab("Post-warmup iteration") # Trace plots
## No divergences to plot.

Same outputs with another prior for sigma: \(\sigma \sim \, {\rm Cauchy} (0,5)\)

data {                                                     
  int<lower=0> N;                                          // number of observations
  int<lower=0> n;                                          // number of levels of the categorical variable
  vector[N] height;                                        // response variables
  matrix[N,n] X;                                           // categorical variable
}

parameters {                                          
  vector[n] beta;                                         // intercepts of the categorical variable
  real sigma;
}

transformed parameters {
vector[N] mu;                                             
mu = X*beta;  
}

model {
  height ~ normal(mu ,sigma);                              // likelihood
  sigma ~ cauchy(0,5);
  beta[1] ~ normal(178,20);
  beta[2] ~ normal(0,10);
  // beta ~ normal(0,1000)
}

generated quantities {
  vector[N] y_rep;
 for(i in 1:N)
 y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_1_2 <- sampling(m5.1.2, data = m5.1.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_1_2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 4c192013905ef0bcb3999ba4bc93cd6e.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## beta[2]      10.79    0.02 0.58   10.03   10.81   11.52  1353 1.00
## sigma         5.55    0.01 0.21    5.29    5.54    5.84  1382 1.00
## mu[1]       160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[2]       149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[3]       149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[4]       160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[5]       149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[6]       160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[7]       149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[8]       160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[9]       149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[10]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[11]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[12]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[13]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[14]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[15]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[16]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[17]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[18]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[19]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[20]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[21]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[22]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[23]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[24]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[25]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[26]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[27]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[28]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[29]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[30]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[31]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[32]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[33]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[34]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[35]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[36]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[37]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[38]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[39]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[40]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[41]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[42]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[43]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[44]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[45]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[46]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[47]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[48]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[49]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[50]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[51]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[52]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[53]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[54]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[55]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[56]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[57]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[58]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[59]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[60]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[61]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[62]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[63]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[64]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[65]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[66]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[67]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[68]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[69]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[70]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[71]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[72]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[73]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[74]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[75]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[76]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[77]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[78]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[79]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[80]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[81]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[82]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[83]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[84]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[85]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[86]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[87]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[88]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[89]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[90]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[91]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[92]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[93]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[94]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[95]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[96]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[97]      149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[98]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[99]      160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[100]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[101]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[102]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[103]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[104]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[105]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[106]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[107]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[108]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[109]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[110]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[111]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[112]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[113]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[114]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[115]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[116]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[117]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[118]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[119]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[120]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[121]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[122]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[123]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[124]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[125]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[126]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[127]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[128]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[129]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[130]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[131]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[132]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[133]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[134]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[135]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[136]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[137]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[138]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[139]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[140]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[141]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[142]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[143]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[144]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[145]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[146]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[147]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[148]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[149]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[150]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[151]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[152]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[153]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[154]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[155]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[156]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[157]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[158]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[159]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[160]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[161]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[162]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[163]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[164]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[165]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[166]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[167]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[168]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[169]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[170]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[171]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[172]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[173]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[174]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[175]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[176]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[177]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[178]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[179]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[180]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[181]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[182]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[183]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[184]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[185]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[186]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[187]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[188]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[189]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[190]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[191]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[192]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[193]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[194]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[195]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[196]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[197]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[198]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[199]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[200]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[201]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[202]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[203]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[204]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[205]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[206]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[207]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[208]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[209]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[210]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[211]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[212]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[213]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[214]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[215]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[216]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[217]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[218]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[219]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[220]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[221]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[222]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[223]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[224]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[225]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[226]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[227]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[228]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[229]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[230]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[231]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[232]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[233]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[234]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[235]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[236]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[237]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[238]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[239]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[240]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[241]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[242]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[243]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[244]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[245]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[246]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[247]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[248]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[249]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[250]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[251]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[252]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[253]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[254]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[255]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[256]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[257]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[258]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[259]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[260]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[261]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[262]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[263]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[264]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[265]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[266]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[267]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[268]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[269]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[270]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[271]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[272]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[273]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[274]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[275]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[276]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[277]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[278]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[279]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[280]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[281]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[282]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[283]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[284]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[285]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[286]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[287]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[288]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[289]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[290]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[291]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[292]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[293]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[294]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[295]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[296]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[297]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[298]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[299]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[300]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[301]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[302]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[303]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[304]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[305]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[306]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[307]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[308]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[309]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[310]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[311]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[312]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[313]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[314]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[315]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[316]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[317]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[318]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[319]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[320]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[321]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[322]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[323]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[324]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[325]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[326]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[327]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[328]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[329]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[330]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[331]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[332]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[333]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[334]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[335]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[336]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[337]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[338]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[339]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[340]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[341]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[342]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[343]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[344]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[345]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[346]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[347]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[348]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[349]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[350]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## mu[351]     149.54    0.01 0.40  149.03  149.55  150.06  1320 1.00
## mu[352]     160.34    0.01 0.43  159.80  160.34  160.88  2106 1.00
## y_rep[1]    160.43    0.13 5.78  153.05  160.49  167.64  1948 1.00
## y_rep[2]    149.69    0.12 5.33  142.86  149.59  156.64  1974 1.00
## y_rep[3]    149.59    0.13 5.81  142.13  149.72  156.97  2072 1.00
## y_rep[4]    160.20    0.15 5.74  152.95  160.13  167.60  1546 1.00
## y_rep[5]    149.47    0.12 5.45  142.47  149.41  156.55  2029 1.00
## y_rep[6]    160.42    0.12 5.55  153.17  160.43  167.55  2041 1.00
## y_rep[7]    149.77    0.13 5.52  142.87  149.73  156.70  1791 1.00
## y_rep[8]    160.31    0.14 5.61  153.08  160.32  167.64  1727 1.00
## y_rep[9]    149.32    0.12 5.65  141.86  149.37  156.49  2053 1.00
## y_rep[10]   160.42    0.12 5.42  153.51  160.27  167.49  1957 1.00
## y_rep[11]   149.60    0.13 5.52  142.54  149.53  156.64  1763 1.00
## y_rep[12]   160.39    0.13 5.66  153.03  160.52  167.66  1986 1.00
## y_rep[13]   149.67    0.14 5.51  142.72  149.63  156.79  1648 1.00
## y_rep[14]   149.76    0.13 5.62  142.61  149.48  157.15  1841 1.00
## y_rep[15]   149.59    0.12 5.47  142.49  149.50  156.37  1943 1.00
## y_rep[16]   160.25    0.12 5.60  153.10  160.26  167.33  2149 1.00
## y_rep[17]   160.51    0.13 5.66  153.08  160.53  167.74  2051 1.00
## y_rep[18]   149.77    0.13 5.60  142.74  149.77  156.98  1995 1.00
## y_rep[19]   160.12    0.12 5.58  153.08  160.07  167.34  2020 1.00
## y_rep[20]   149.64    0.12 5.47  142.84  149.59  156.65  1947 1.00
## y_rep[21]   160.52    0.12 5.72  152.95  160.45  168.10  2110 1.00
## y_rep[22]   149.64    0.13 5.64  142.55  149.39  156.82  1770 1.00
## y_rep[23]   149.60    0.13 5.67  142.45  149.50  156.62  2044 1.00
## y_rep[24]   160.13    0.12 5.68  152.79  160.13  167.26  2136 1.00
## y_rep[25]   160.48    0.13 5.64  153.08  160.67  167.60  2006 1.00
## y_rep[26]   149.42    0.13 5.59  142.22  149.53  156.56  1938 1.00
## y_rep[27]   149.60    0.12 5.63  142.29  149.68  156.73  2028 1.00
## y_rep[28]   149.50    0.13 5.65  142.32  149.62  156.96  1853 1.00
## y_rep[29]   149.43    0.12 5.43  142.27  149.41  156.38  1938 1.00
## y_rep[30]   149.72    0.13 5.46  142.69  149.68  156.71  1799 1.00
## y_rep[31]   149.54    0.12 5.47  142.45  149.67  156.27  1953 1.00
## y_rep[32]   160.37    0.13 5.50  153.20  160.38  167.46  1924 1.00
## y_rep[33]   160.20    0.13 5.71  152.81  160.10  167.48  1950 1.00
## y_rep[34]   160.28    0.13 5.66  153.05  160.17  167.57  2001 1.00
## y_rep[35]   160.29    0.12 5.43  153.39  160.20  167.11  1951 1.00
## y_rep[36]   149.57    0.12 5.51  142.68  149.60  156.50  2042 1.00
## y_rep[37]   149.33    0.13 5.55  142.29  149.32  156.60  1829 1.00
## y_rep[38]   160.31    0.13 5.60  153.13  160.30  167.36  1852 1.00
## y_rep[39]   149.47    0.13 5.59  142.27  149.27  156.77  1871 1.00
## y_rep[40]   160.25    0.12 5.63  153.25  160.23  167.28  2036 1.00
## y_rep[41]   160.10    0.12 5.64  152.98  160.14  167.36  2130 1.00
## y_rep[42]   160.42    0.12 5.39  153.40  160.57  167.11  1947 1.00
## y_rep[43]   149.53    0.15 5.63  142.35  149.51  156.61  1451 1.00
## y_rep[44]   149.52    0.13 5.52  142.73  149.64  156.30  1930 1.00
## y_rep[45]   149.71    0.13 5.70  142.50  149.78  157.06  2061 1.00
## y_rep[46]   149.34    0.12 5.50  142.21  149.39  156.18  1979 1.00
## y_rep[47]   160.47    0.13 5.47  153.52  160.45  167.43  1913 1.00
## y_rep[48]   149.58    0.12 5.44  142.91  149.60  156.72  2051 1.00
## y_rep[49]   160.37    0.12 5.70  152.99  160.37  167.88  2180 1.00
## y_rep[50]   149.58    0.13 5.67  142.34  149.68  156.57  2008 1.00
## y_rep[51]   149.14    0.13 5.64  141.86  149.26  156.45  1956 1.00
## y_rep[52]   149.78    0.13 5.68  142.25  149.88  157.02  1934 1.00
## y_rep[53]   160.31    0.12 5.60  153.29  160.23  167.65  2185 1.00
## y_rep[54]   160.15    0.13 5.70  152.95  160.19  167.25  1919 1.00
## y_rep[55]   149.42    0.13 5.51  142.38  149.43  156.80  1917 1.00
## y_rep[56]   160.40    0.12 5.53  153.19  160.41  167.47  2002 1.00
## y_rep[57]   149.40    0.13 5.70  142.17  149.38  156.47  1983 1.00
## y_rep[58]   160.34    0.13 5.64  153.20  160.46  167.51  1932 1.00
## y_rep[59]   149.84    0.12 5.56  142.62  149.85  156.81  1995 1.00
## y_rep[60]   160.19    0.14 5.66  152.95  160.16  167.32  1711 1.00
## y_rep[61]   149.50    0.12 5.55  142.43  149.45  156.69  2011 1.00
## y_rep[62]   149.63    0.12 5.40  142.72  149.64  156.74  2123 1.00
## y_rep[63]   160.38    0.12 5.45  153.51  160.21  167.38  2043 1.00
## y_rep[64]   160.47    0.12 5.54  153.31  160.42  167.53  2059 1.00
## y_rep[65]   149.34    0.14 5.70  141.92  149.45  156.53  1732 1.00
## y_rep[66]   160.43    0.13 5.68  153.23  160.56  167.76  2012 1.00
## y_rep[67]   149.55    0.14 5.63  142.33  149.62  156.81  1625 1.00
## y_rep[68]   149.48    0.12 5.47  142.39  149.44  156.43  2003 1.00
## y_rep[69]   160.42    0.13 5.60  152.88  160.56  167.52  2005 1.00
## y_rep[70]   149.67    0.13 5.61  142.46  149.55  156.85  1862 1.00
## y_rep[71]   160.43    0.12 5.54  153.39  160.50  167.46  2097 1.00
## y_rep[72]   160.41    0.12 5.70  153.08  160.33  167.75  2082 1.00
## y_rep[73]   160.48    0.13 5.54  153.22  160.62  167.44  1928 1.00
## y_rep[74]   149.45    0.12 5.57  142.20  149.43  156.66  2073 1.00
## y_rep[75]   149.34    0.12 5.50  142.33  149.38  156.42  2011 1.00
## y_rep[76]   149.45    0.12 5.44  142.46  149.40  156.61  1914 1.00
## y_rep[77]   149.56    0.12 5.56  142.66  149.57  156.76  2132 1.00
## y_rep[78]   160.23    0.12 5.51  153.23  160.11  167.36  1950 1.00
## y_rep[79]   149.49    0.12 5.47  142.68  149.60  156.36  2174 1.00
## y_rep[80]   149.70    0.13 5.62  142.54  149.63  156.93  1843 1.00
## y_rep[81]   160.40    0.13 5.59  153.03  160.44  167.66  1964 1.00
## y_rep[82]   160.34    0.12 5.51  153.03  160.54  167.30  1963 1.00
## y_rep[83]   149.53    0.12 5.54  142.18  149.55  156.74  2009 1.00
## y_rep[84]   160.61    0.12 5.51  153.64  160.56  167.80  2108 1.00
## y_rep[85]   149.66    0.12 5.61  142.32  149.73  156.63  2023 1.00
## y_rep[86]   160.51    0.12 5.51  153.42  160.70  167.50  2136 1.00
## y_rep[87]   149.73    0.14 5.68  142.38  149.78  157.10  1691 1.00
## y_rep[88]   160.31    0.12 5.49  153.21  160.31  167.11  2046 1.00
## y_rep[89]   149.44    0.13 5.58  142.25  149.52  156.64  1885 1.00
## y_rep[90]   160.12    0.12 5.52  152.99  160.20  167.08  2028 1.00
## y_rep[91]   160.30    0.12 5.63  153.31  160.15  167.89  2101 1.00
## y_rep[92]   160.19    0.13 5.56  152.98  160.22  167.40  1782 1.00
## y_rep[93]   149.29    0.12 5.50  142.04  149.37  156.42  2001 1.00
## y_rep[94]   149.57    0.13 5.66  142.65  149.44  156.73  1872 1.00
## y_rep[95]   160.36    0.12 5.52  153.32  160.31  167.62  1967 1.00
## y_rep[96]   160.22    0.12 5.59  152.88  160.19  167.38  2077 1.00
## y_rep[97]   149.71    0.13 5.73  142.37  149.77  157.13  2084 1.00
## y_rep[98]   160.26    0.12 5.56  153.02  160.25  167.49  1997 1.00
## y_rep[99]   160.37    0.12 5.53  153.35  160.27  167.40  2041 1.00
## y_rep[100]  149.42    0.13 5.68  142.07  149.55  156.53  1811 1.00
## y_rep[101]  160.33    0.12 5.48  153.35  160.39  167.27  2262 1.00
## y_rep[102]  149.61    0.13 5.52  142.57  149.65  156.56  1830 1.00
## y_rep[103]  149.77    0.12 5.43  142.58  149.73  156.82  2176 1.00
## y_rep[104]  160.44    0.13 5.55  153.61  160.34  167.64  1969 1.00
## y_rep[105]  160.27    0.12 5.61  153.05  160.30  167.39  2115 1.00
## y_rep[106]  149.61    0.12 5.57  142.67  149.64  156.65  2099 1.00
## y_rep[107]  149.61    0.13 5.59  142.65  149.52  156.60  1960 1.00
## y_rep[108]  149.39    0.13 5.64  142.11  149.47  156.72  1936 1.00
## y_rep[109]  160.48    0.13 5.51  153.70  160.37  167.69  1924 1.00
## y_rep[110]  149.56    0.13 5.66  142.34  149.58  156.91  2028 1.00
## y_rep[111]  160.20    0.13 5.53  153.26  160.04  167.21  1855 1.00
## y_rep[112]  149.46    0.13 5.69  142.21  149.43  156.68  1853 1.00
## y_rep[113]  149.53    0.13 5.68  142.36  149.62  156.78  1971 1.00
## y_rep[114]  160.41    0.13 5.63  153.13  160.40  167.45  1995 1.00
## y_rep[115]  160.40    0.13 5.68  152.96  160.63  167.59  1966 1.00
## y_rep[116]  149.48    0.13 5.69  142.18  149.42  156.79  1881 1.00
## y_rep[117]  160.59    0.13 5.52  153.51  160.60  167.51  1926 1.00
## y_rep[118]  160.35    0.12 5.44  153.56  160.31  167.31  1952 1.00
## y_rep[119]  149.62    0.12 5.59  142.45  149.61  156.63  2051 1.00
## y_rep[120]  149.49    0.13 5.65  142.34  149.41  156.81  2009 1.00
## y_rep[121]  160.31    0.12 5.67  153.17  160.47  167.59  2068 1.00
## y_rep[122]  149.56    0.12 5.64  142.06  149.76  156.75  2060 1.00
## y_rep[123]  160.45    0.12 5.30  153.81  160.48  167.29  2067 1.00
## y_rep[124]  149.79    0.12 5.48  142.81  149.79  156.88  1947 1.00
## y_rep[125]  149.44    0.12 5.66  142.17  149.45  156.73  2141 1.00
## y_rep[126]  149.39    0.13 5.61  142.15  149.46  156.59  1859 1.00
## y_rep[127]  149.36    0.12 5.48  142.54  149.22  156.45  2072 1.00
## y_rep[128]  160.46    0.13 5.67  153.08  160.60  167.61  1937 1.00
## y_rep[129]  149.81    0.12 5.56  142.65  149.71  157.04  2167 1.00
## y_rep[130]  149.37    0.13 5.64  142.00  149.49  156.43  2022 1.00
## y_rep[131]  160.38    0.13 5.51  153.58  160.32  167.50  1944 1.00
## y_rep[132]  160.30    0.12 5.52  153.29  160.24  167.33  2023 1.00
## y_rep[133]  149.56    0.13 5.60  142.32  149.60  156.86  1881 1.00
## y_rep[134]  160.30    0.13 5.56  152.97  160.38  167.24  1790 1.00
## y_rep[135]  149.51    0.12 5.47  142.61  149.45  156.54  1999 1.00
## y_rep[136]  160.19    0.12 5.49  153.14  160.16  167.38  2029 1.00
## y_rep[137]  160.38    0.12 5.53  153.26  160.43  167.49  1985 1.00
## y_rep[138]  160.20    0.13 5.61  153.00  159.99  167.47  1752 1.00
## y_rep[139]  149.48    0.12 5.56  142.36  149.38  156.70  2040 1.00
## y_rep[140]  160.69    0.12 5.66  153.37  160.71  168.04  2100 1.00
## y_rep[141]  149.66    0.13 5.41  142.60  149.74  156.51  1832 1.00
## y_rep[142]  149.81    0.12 5.56  142.65  149.82  156.86  2023 1.00
## y_rep[143]  149.47    0.12 5.51  142.47  149.47  156.53  2003 1.00
## y_rep[144]  149.45    0.12 5.35  142.74  149.44  156.29  1836 1.00
## y_rep[145]  149.47    0.13 5.44  142.61  149.40  156.42  1777 1.00
## y_rep[146]  160.24    0.12 5.64  153.16  160.07  167.62  2051 1.00
## y_rep[147]  149.49    0.13 5.55  142.44  149.44  156.68  1876 1.00
## y_rep[148]  149.65    0.12 5.45  142.70  149.60  156.86  1953 1.00
## y_rep[149]  160.40    0.13 5.66  153.09  160.40  167.59  1967 1.00
## y_rep[150]  149.74    0.13 5.71  142.25  149.66  157.20  1838 1.00
## y_rep[151]  160.36    0.13 5.58  153.25  160.54  167.38  1792 1.00
## y_rep[152]  149.52    0.14 5.62  142.57  149.36  156.80  1607 1.00
## y_rep[153]  160.33    0.13 5.52  153.40  160.27  167.25  1947 1.00
## y_rep[154]  149.49    0.13 5.44  142.44  149.58  156.48  1827 1.00
## y_rep[155]  160.48    0.12 5.53  153.46  160.68  167.71  2007 1.00
## y_rep[156]  149.66    0.13 5.54  142.68  149.80  156.70  1754 1.00
## y_rep[157]  149.32    0.12 5.55  142.18  149.25  156.29  2070 1.00
## y_rep[158]  160.44    0.12 5.44  153.43  160.53  167.47  1944 1.00
## y_rep[159]  160.16    0.13 5.60  153.31  160.07  167.33  1993 1.00
## y_rep[160]  149.43    0.13 5.41  142.69  149.40  156.25  1824 1.00
## y_rep[161]  160.37    0.13 5.53  153.16  160.43  167.37  1890 1.00
## y_rep[162]  149.65    0.12 5.50  142.90  149.61  156.75  1992 1.00
## y_rep[163]  160.38    0.13 5.55  153.28  160.34  167.44  1935 1.00
## y_rep[164]  160.51    0.12 5.60  153.25  160.46  167.92  2175 1.00
## y_rep[165]  160.22    0.13 5.66  152.92  160.07  167.48  1827 1.00
## y_rep[166]  149.56    0.12 5.66  142.34  149.53  156.93  2100 1.00
## y_rep[167]  160.49    0.12 5.51  153.43  160.34  167.43  2041 1.00
## y_rep[168]  149.63    0.12 5.62  142.59  149.61  156.80  2111 1.00
## y_rep[169]  149.48    0.13 5.67  142.16  149.54  156.42  1984 1.00
## y_rep[170]  149.74    0.12 5.49  142.75  149.70  156.77  2063 1.00
## y_rep[171]  149.59    0.13 5.70  142.28  149.67  156.77  1909 1.00
## y_rep[172]  160.38    0.13 5.48  153.20  160.42  167.30  1833 1.00
## y_rep[173]  149.50    0.12 5.65  142.48  149.49  156.56  2153 1.00
## y_rep[174]  160.32    0.13 5.67  153.19  160.23  167.81  1864 1.00
## y_rep[175]  149.44    0.12 5.37  142.47  149.30  156.38  2104 1.00
## y_rep[176]  160.25    0.13 5.52  153.29  160.21  167.15  1696 1.00
## y_rep[177]  149.42    0.12 5.45  142.40  149.44  156.41  1980 1.00
## y_rep[178]  149.64    0.13 5.60  142.73  149.58  156.72  1994 1.00
## y_rep[179]  160.20    0.12 5.55  153.20  160.25  167.15  2003 1.00
## y_rep[180]  149.50    0.13 5.48  142.61  149.50  156.55  1922 1.00
## y_rep[181]  160.10    0.14 5.63  152.84  160.07  167.53  1733 1.00
## y_rep[182]  149.48    0.12 5.71  142.13  149.62  156.84  2178 1.00
## y_rep[183]  149.33    0.13 5.67  142.20  149.26  156.51  2047 1.00
## y_rep[184]  149.43    0.12 5.56  142.28  149.47  156.57  2033 1.00
## y_rep[185]  160.33    0.12 5.74  153.06  160.24  167.80  2124 1.00
## y_rep[186]  160.50    0.13 5.67  153.41  160.56  167.72  1997 1.00
## y_rep[187]  149.24    0.12 5.60  142.15  149.31  156.34  2096 1.00
## y_rep[188]  160.41    0.12 5.54  153.34  160.36  167.48  1972 1.00
## y_rep[189]  149.46    0.13 5.61  141.86  149.65  156.42  1938 1.00
## y_rep[190]  160.42    0.13 5.57  153.03  160.50  167.52  1880 1.00
## y_rep[191]  149.74    0.12 5.50  142.69  149.71  156.76  1983 1.01
## y_rep[192]  160.52    0.14 5.57  153.51  160.46  167.69  1633 1.00
## y_rep[193]  149.45    0.13 5.60  142.43  149.38  156.64  2000 1.00
## y_rep[194]  160.22    0.12 5.44  153.17  160.25  167.13  1993 1.00
## y_rep[195]  149.56    0.13 5.63  142.55  149.50  156.87  1894 1.00
## y_rep[196]  149.53    0.13 5.76  141.95  149.54  156.92  1866 1.00
## y_rep[197]  149.50    0.13 5.74  142.03  149.67  156.85  1914 1.00
## y_rep[198]  160.30    0.12 5.58  153.18  160.19  167.55  2176 1.00
## y_rep[199]  160.45    0.12 5.49  153.23  160.51  167.65  2029 1.00
## y_rep[200]  149.42    0.12 5.60  142.13  149.54  156.59  2114 1.00
## y_rep[201]  160.25    0.12 5.43  153.46  160.26  167.27  1988 1.00
## y_rep[202]  149.66    0.13 5.59  142.24  149.69  156.77  1971 1.00
## y_rep[203]  160.57    0.13 5.71  153.18  160.51  167.99  1887 1.00
## y_rep[204]  149.44    0.13 5.54  142.33  149.54  156.24  1901 1.00
## y_rep[205]  160.56    0.12 5.54  153.30  160.73  167.50  2141 1.00
## y_rep[206]  149.68    0.12 5.57  142.62  149.61  156.88  1994 1.00
## y_rep[207]  149.57    0.13 5.56  142.50  149.52  156.55  1901 1.00
## y_rep[208]  149.64    0.13 5.52  142.75  149.55  156.64  1943 1.00
## y_rep[209]  149.39    0.13 5.59  142.14  149.31  156.51  1945 1.00
## y_rep[210]  149.45    0.12 5.47  142.47  149.56  156.33  1982 1.00
## y_rep[211]  160.54    0.13 5.57  153.36  160.53  167.64  1743 1.00
## y_rep[212]  160.57    0.12 5.51  153.54  160.65  167.51  2123 1.00
## y_rep[213]  160.51    0.13 5.58  153.41  160.48  167.56  1979 1.00
## y_rep[214]  149.62    0.13 5.64  142.23  149.70  156.81  1890 1.00
## y_rep[215]  160.48    0.12 5.58  153.24  160.57  167.26  2001 1.00
## y_rep[216]  160.29    0.13 5.64  153.33  160.18  167.81  1957 1.00
## y_rep[217]  160.34    0.13 5.63  153.30  160.33  167.61  1841 1.00
## y_rep[218]  149.65    0.13 5.64  142.58  149.64  156.78  1811 1.00
## y_rep[219]  160.53    0.12 5.54  153.33  160.48  167.63  2000 1.00
## y_rep[220]  149.48    0.13 5.53  142.40  149.41  156.61  1897 1.00
## y_rep[221]  149.78    0.12 5.56  142.76  149.61  157.07  2099 1.00
## y_rep[222]  149.63    0.13 5.59  142.52  149.65  156.60  1846 1.00
## y_rep[223]  160.31    0.13 5.43  152.82  160.48  167.38  1800 1.00
## y_rep[224]  149.24    0.13 5.66  142.01  149.28  156.70  2005 1.00
## y_rep[225]  160.27    0.13 5.57  152.93  160.38  167.22  1892 1.00
## y_rep[226]  149.59    0.13 5.72  142.23  149.80  156.81  1989 1.00
## y_rep[227]  149.38    0.13 5.61  142.24  149.39  156.65  1778 1.00
## y_rep[228]  160.61    0.12 5.44  153.60  160.65  167.55  2047 1.00
## y_rep[229]  149.42    0.14 5.71  142.20  149.45  157.04  1598 1.00
## y_rep[230]  160.54    0.12 5.49  153.66  160.51  167.60  1971 1.00
## y_rep[231]  160.43    0.12 5.59  153.33  160.21  168.03  2251 1.00
## y_rep[232]  149.53    0.13 5.66  142.17  149.44  156.79  1782 1.00
## y_rep[233]  149.49    0.12 5.41  142.68  149.61  156.40  1994 1.00
## y_rep[234]  160.34    0.12 5.43  153.36  160.33  167.17  2153 1.00
## y_rep[235]  160.23    0.12 5.64  152.91  160.21  167.47  2116 1.00
## y_rep[236]  149.43    0.12 5.38  142.55  149.46  156.39  2046 1.00
## y_rep[237]  160.00    0.12 5.55  152.95  159.91  166.94  2114 1.00
## y_rep[238]  149.49    0.13 5.50  142.58  149.47  156.44  1912 1.00
## y_rep[239]  149.42    0.13 5.63  141.99  149.50  156.71  1961 1.00
## y_rep[240]  149.42    0.13 5.67  141.99  149.62  156.56  2037 1.00
## y_rep[241]  149.59    0.13 5.50  142.46  149.74  156.58  1922 1.00
## y_rep[242]  160.41    0.13 5.71  153.32  160.22  167.72  1962 1.00
## y_rep[243]  160.33    0.13 5.65  153.11  160.48  167.34  1849 1.00
## y_rep[244]  149.52    0.15 5.64  142.32  149.50  156.84  1451 1.00
## y_rep[245]  160.37    0.12 5.57  153.17  160.49  167.47  2047 1.00
## y_rep[246]  149.54    0.13 5.50  142.42  149.53  156.44  1904 1.00
## y_rep[247]  160.29    0.13 5.69  153.16  160.08  167.54  1841 1.00
## y_rep[248]  149.55    0.12 5.58  142.44  149.50  156.72  2057 1.00
## y_rep[249]  160.38    0.12 5.47  153.48  160.48  167.23  1937 1.00
## y_rep[250]  160.37    0.12 5.57  153.45  160.17  167.57  2063 1.00
## y_rep[251]  160.40    0.13 5.58  153.41  160.42  167.66  1897 1.00
## y_rep[252]  160.36    0.13 5.64  153.42  160.27  167.60  1880 1.00
## y_rep[253]  149.63    0.12 5.40  142.62  149.63  156.56  2100 1.00
## y_rep[254]  149.45    0.13 5.63  142.29  149.36  156.86  1917 1.00
## y_rep[255]  149.45    0.13 5.60  142.31  149.52  156.59  2005 1.00
## y_rep[256]  149.62    0.13 5.39  142.50  149.63  156.50  1839 1.00
## y_rep[257]  160.54    0.12 5.55  153.57  160.40  167.80  2180 1.00
## y_rep[258]  149.65    0.13 5.59  142.71  149.59  156.80  1742 1.00
## y_rep[259]  160.38    0.13 5.54  153.20  160.36  167.45  1825 1.00
## y_rep[260]  160.28    0.12 5.64  152.94  160.24  167.60  2136 1.00
## y_rep[261]  160.04    0.13 5.59  152.75  160.12  167.24  1931 1.00
## y_rep[262]  149.75    0.13 5.79  142.34  149.81  157.00  2026 1.00
## y_rep[263]  149.66    0.13 5.78  141.93  149.87  156.78  1953 1.00
## y_rep[264]  149.41    0.12 5.56  142.61  149.40  156.41  2029 1.00
## y_rep[265]  160.20    0.13 5.58  153.08  160.20  167.20  1893 1.00
## y_rep[266]  149.41    0.13 5.55  142.44  149.31  156.45  1891 1.00
## y_rep[267]  160.24    0.12 5.54  152.94  160.40  167.15  1974 1.00
## y_rep[268]  149.33    0.12 5.47  142.30  149.35  156.63  2024 1.00
## y_rep[269]  149.68    0.12 5.52  142.28  149.83  156.67  2040 1.00
## y_rep[270]  160.33    0.12 5.57  153.10  160.28  167.41  2058 1.00
## y_rep[271]  160.44    0.12 5.50  153.43  160.49  167.54  2121 1.00
## y_rep[272]  149.68    0.12 5.54  142.61  149.79  156.69  1995 1.00
## y_rep[273]  160.32    0.13 5.52  152.99  160.35  167.45  1757 1.00
## y_rep[274]  160.26    0.12 5.50  153.26  160.26  167.30  2092 1.00
## y_rep[275]  149.30    0.13 5.76  141.93  149.17  156.93  2085 1.00
## y_rep[276]  160.55    0.12 5.53  153.33  160.66  167.49  2107 1.00
## y_rep[277]  149.50    0.13 5.63  142.21  149.69  156.74  2004 1.00
## y_rep[278]  149.43    0.13 5.49  142.56  149.50  156.42  1709 1.00
## y_rep[279]  160.35    0.12 5.54  153.29  160.40  167.32  1992 1.00
## y_rep[280]  149.55    0.13 5.51  142.67  149.38  156.58  1931 1.00
## y_rep[281]  149.47    0.13 5.45  142.40  149.64  156.36  1775 1.00
## y_rep[282]  160.29    0.13 5.57  153.27  160.28  167.37  1774 1.00
## y_rep[283]  149.65    0.12 5.43  142.67  149.62  156.85  1987 1.00
## y_rep[284]  160.19    0.13 5.69  152.85  160.16  167.56  2071 1.00
## y_rep[285]  149.75    0.12 5.63  142.83  149.63  156.89  2061 1.00
## y_rep[286]  160.61    0.12 5.59  153.50  160.65  167.97  2064 1.00
## y_rep[287]  160.30    0.13 5.73  153.10  160.31  167.67  1935 1.00
## y_rep[288]  160.52    0.13 5.57  153.25  160.56  167.74  1860 1.00
## y_rep[289]  149.43    0.12 5.64  142.46  149.27  156.61  2127 1.00
## y_rep[290]  160.28    0.13 5.74  152.83  160.31  167.68  2035 1.00
## y_rep[291]  160.41    0.12 5.48  153.33  160.49  167.32  1976 1.00
## y_rep[292]  160.38    0.13 5.54  153.28  160.30  167.52  1749 1.00
## y_rep[293]  160.26    0.12 5.62  153.10  160.34  167.36  2061 1.00
## y_rep[294]  149.63    0.13 5.51  142.62  149.54  156.64  1885 1.00
## y_rep[295]  160.15    0.13 5.56  153.11  160.16  167.33  1822 1.00
## y_rep[296]  149.41    0.12 5.47  142.54  149.17  156.66  1986 1.00
## y_rep[297]  149.64    0.13 5.74  142.16  149.74  157.23  1834 1.00
## y_rep[298]  149.29    0.13 5.63  142.05  149.19  156.68  1873 1.00
## y_rep[299]  160.33    0.13 5.70  153.05  160.37  167.61  1785 1.00
## y_rep[300]  160.41    0.13 5.64  152.97  160.48  167.47  1952 1.00
## y_rep[301]  149.59    0.13 5.47  142.55  149.63  156.52  1914 1.00
## y_rep[302]  160.46    0.13 5.73  153.06  160.40  167.59  2074 1.00
## y_rep[303]  149.65    0.12 5.54  142.74  149.59  156.83  2053 1.00
## y_rep[304]  160.36    0.12 5.57  153.38  160.28  167.82  2255 1.00
## y_rep[305]  160.35    0.13 5.55  153.18  160.50  167.48  1950 1.00
## y_rep[306]  160.49    0.13 5.50  153.26  160.48  167.59  1936 1.00
## y_rep[307]  160.49    0.13 5.47  153.69  160.65  167.31  1851 1.00
## y_rep[308]  149.48    0.13 5.64  142.27  149.34  156.61  1926 1.00
## y_rep[309]  149.62    0.13 5.60  142.56  149.47  156.81  1862 1.00
## y_rep[310]  149.39    0.12 5.64  142.05  149.41  156.76  2070 1.00
## y_rep[311]  149.60    0.14 5.61  142.24  149.67  156.67  1694 1.00
## y_rep[312]  149.59    0.13 5.63  142.31  149.60  157.09  1992 1.00
## y_rep[313]  160.43    0.14 5.72  153.05  160.51  167.74  1693 1.00
## y_rep[314]  149.40    0.13 5.52  142.01  149.59  156.30  1756 1.00
## y_rep[315]  160.42    0.12 5.52  153.27  160.53  167.55  2001 1.00
## y_rep[316]  149.61    0.13 5.63  142.49  149.75  156.97  1967 1.00
## y_rep[317]  160.15    0.13 5.78  153.05  160.05  167.58  1986 1.00
## y_rep[318]  149.51    0.12 5.43  142.54  149.59  156.43  2131 1.00
## y_rep[319]  160.24    0.12 5.49  153.21  160.35  167.05  2003 1.00
## y_rep[320]  149.54    0.12 5.64  142.44  149.56  156.82  2095 1.00
## y_rep[321]  160.17    0.13 5.66  152.89  160.11  167.40  1950 1.00
## y_rep[322]  149.28    0.13 5.45  142.40  149.11  156.42  1863 1.00
## y_rep[323]  160.34    0.12 5.47  153.26  160.35  167.43  2090 1.00
## y_rep[324]  160.20    0.13 5.67  152.88  160.11  167.43  1923 1.00
## y_rep[325]  149.65    0.13 5.59  142.49  149.65  156.95  1826 1.00
## y_rep[326]  149.76    0.12 5.50  142.68  149.75  156.89  2090 1.00
## y_rep[327]  149.42    0.13 5.64  142.31  149.44  156.51  1902 1.00
## y_rep[328]  160.26    0.13 5.58  153.26  160.27  167.26  1852 1.00
## y_rep[329]  160.53    0.12 5.66  153.33  160.51  167.79  2085 1.00
## y_rep[330]  160.32    0.12 5.50  153.34  160.31  167.48  1984 1.00
## y_rep[331]  160.17    0.13 5.64  152.82  160.35  167.37  1885 1.00
## y_rep[332]  160.26    0.12 5.58  153.11  160.25  167.41  2037 1.00
## y_rep[333]  149.52    0.13 5.63  142.32  149.61  156.57  1956 1.00
## y_rep[334]  160.17    0.12 5.48  153.07  160.29  167.15  2005 1.00
## y_rep[335]  149.37    0.12 5.45  142.26  149.48  156.30  1943 1.00
## y_rep[336]  160.37    0.13 5.45  153.41  160.48  167.12  1891 1.00
## y_rep[337]  149.65    0.12 5.42  142.52  149.66  156.76  2052 1.00
## y_rep[338]  149.30    0.12 5.62  142.06  149.27  156.39  2048 1.00
## y_rep[339]  149.63    0.13 5.60  142.68  149.47  156.84  1988 1.00
## y_rep[340]  149.48    0.12 5.56  142.35  149.55  156.53  2068 1.00
## y_rep[341]  149.56    0.13 5.54  142.56  149.60  156.62  1730 1.00
## y_rep[342]  160.47    0.13 5.66  153.32  160.51  167.78  2051 1.00
## y_rep[343]  149.39    0.13 5.56  142.26  149.37  156.47  1851 1.00
## y_rep[344]  160.53    0.13 5.42  153.54  160.56  167.47  1875 1.00
## y_rep[345]  149.71    0.12 5.55  142.59  149.59  156.75  2008 1.00
## y_rep[346]  160.35    0.12 5.45  153.32  160.35  167.40  1946 1.00
## y_rep[347]  149.64    0.13 5.54  142.54  149.72  156.69  1820 1.00
## y_rep[348]  149.50    0.12 5.53  142.56  149.37  156.48  2025 1.00
## y_rep[349]  149.81    0.13 5.56  142.81  149.83  156.94  1738 1.00
## y_rep[350]  160.21    0.13 5.51  153.06  160.24  167.28  1878 1.00
## y_rep[351]  149.56    0.12 5.60  142.14  149.47  157.02  2074 1.00
## y_rep[352]  160.36    0.12 5.49  153.24  160.45  167.53  2041 1.00
## lp__       -781.30    0.04 1.18 -782.94 -781.00 -780.11   902 1.00
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 15:25:44 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

4.2 Index variable

Here we consider the variable male as an index variable, with value 1 whenever the person is a female and a value 2 whenever the person is a male.

howell2$male.index <- ifelse(howell2$male==1,2,1)
m5.2.data <- list(N = nrow(howell2), height = howell2$height, male = as.integer(howell2$male.index), n = length(unique(howell2$male.index)))
str(m5.2.data)
## List of 4
##  $ N     : int 352
##  $ height: num [1:352] 152 140 137 157 145 ...
##  $ male  : int [1:352] 2 1 1 2 1 2 1 2 1 2 ...
##  $ n     : int 2

Model equation:

\[\begin{align*} h_{i} &\sim \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} &= \alpha_{\text{SEX}[\textit{i}]}\\[4pt] \alpha_{j} &\sim \mathcal{N}(178,20) \quad \text{, for} \, j = 1..2\\[4pt] \sigma^2 &\sim \mathcal{U}(0,5)\\[4pt] \end{align*}\]
data {                                                     
  int<lower=0> N;                                          // number of observations
  int<lower=0> n;                                          // number of levels of the categorical variable
  vector[N] height;                                        // response variables
  int<lower=1,upper=n> male[N];                            // categorical variable
}

parameters {                                          
  vector[n] beta;    // real beta[n];  (idem)               // intercepts of the categorical variable
  real sigma;
}

transformed parameters {
vector[N] mu;                                              // create one variable with predictions for each obs
for (i in 1:N)                                             // loop for all cases 
mu[i] = beta[male[i]];  
}

model {
  height ~ normal(mu ,sigma);                              // likelihood
  sigma ~ cauchy(0,5);
}

generated quantities {
  vector[N] y_rep;

 for(i in 1:N)
 y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_2 <- sampling(m5.2, data = m5.2.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_2, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 0da8a26b283bfab1846d7d15060893cd.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## beta[2]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## sigma         5.54    0.01 0.22    5.27    5.53    5.82  1698 1.00
## mu[1]       160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[2]       149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[3]       149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[4]       160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[5]       149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[6]       160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[7]       149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[8]       160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[9]       149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[10]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[11]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[12]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[13]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[14]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[15]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[16]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[17]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[18]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[19]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[20]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[21]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[22]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[23]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[24]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[25]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[26]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[27]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[28]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[29]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[30]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[31]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[32]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[33]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[34]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[35]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[36]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[37]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[38]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[39]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[40]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[41]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[42]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[43]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[44]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[45]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[46]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[47]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[48]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[49]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[50]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[51]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[52]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[53]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[54]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[55]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[56]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[57]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[58]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[59]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[60]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[61]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[62]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[63]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[64]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[65]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[66]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[67]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[68]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[69]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[70]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[71]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[72]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[73]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[74]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[75]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[76]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[77]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[78]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[79]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[80]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[81]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[82]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[83]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[84]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[85]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[86]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[87]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[88]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[89]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[90]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[91]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[92]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[93]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[94]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[95]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[96]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[97]      149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[98]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[99]      160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[100]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[101]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[102]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[103]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[104]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[105]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[106]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[107]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[108]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[109]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[110]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[111]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[112]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[113]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[114]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[115]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[116]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[117]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[118]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[119]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[120]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[121]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[122]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[123]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[124]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[125]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[126]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[127]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[128]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[129]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[130]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[131]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[132]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[133]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[134]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[135]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[136]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[137]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[138]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[139]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[140]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[141]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[142]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[143]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[144]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[145]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[146]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[147]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[148]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[149]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[150]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[151]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[152]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[153]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[154]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[155]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[156]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[157]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[158]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[159]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[160]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[161]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[162]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[163]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[164]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[165]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[166]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[167]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[168]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[169]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[170]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[171]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[172]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[173]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[174]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[175]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[176]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[177]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[178]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[179]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[180]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[181]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[182]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[183]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[184]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[185]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[186]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[187]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[188]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[189]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[190]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[191]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[192]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[193]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[194]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[195]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[196]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[197]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[198]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[199]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[200]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[201]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[202]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[203]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[204]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[205]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[206]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[207]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[208]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[209]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[210]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[211]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[212]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[213]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[214]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[215]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[216]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[217]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[218]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[219]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[220]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[221]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[222]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[223]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[224]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[225]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[226]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[227]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[228]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[229]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[230]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[231]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[232]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[233]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[234]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[235]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[236]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[237]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[238]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[239]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[240]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[241]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[242]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[243]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[244]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[245]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[246]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[247]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[248]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[249]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[250]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[251]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[252]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[253]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[254]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[255]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[256]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[257]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[258]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[259]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[260]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[261]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[262]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[263]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[264]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[265]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[266]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[267]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[268]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[269]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[270]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[271]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[272]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[273]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[274]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[275]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[276]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[277]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[278]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[279]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[280]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[281]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[282]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[283]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[284]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[285]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[286]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[287]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[288]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[289]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[290]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[291]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[292]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[293]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[294]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[295]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[296]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[297]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[298]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[299]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[300]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[301]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[302]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[303]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[304]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[305]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[306]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[307]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[308]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[309]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[310]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[311]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[312]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[313]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[314]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[315]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[316]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[317]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[318]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[319]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[320]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[321]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[322]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[323]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[324]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[325]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[326]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[327]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[328]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[329]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[330]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[331]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[332]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[333]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[334]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[335]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[336]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[337]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[338]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[339]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[340]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[341]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[342]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[343]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[344]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[345]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[346]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[347]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[348]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[349]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[350]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## mu[351]     149.51    0.01 0.41  149.00  149.51  150.02  1769 1.00
## mu[352]     160.35    0.01 0.43  159.80  160.36  160.91  1858 1.00
## y_rep[1]    160.34    0.13 5.61  153.07  160.38  167.35  1934 1.00
## y_rep[2]    149.47    0.12 5.57  142.36  149.49  156.57  2073 1.00
## y_rep[3]    149.60    0.13 5.62  142.32  149.68  156.99  1829 1.00
## y_rep[4]    160.39    0.13 5.74  153.12  160.45  167.75  2057 1.00
## y_rep[5]    149.44    0.12 5.56  142.56  149.42  156.47  2058 1.00
## y_rep[6]    160.39    0.13 5.51  153.49  160.41  167.42  1796 1.00
## y_rep[7]    149.27    0.13 5.53  142.26  149.17  156.37  1792 1.00
## y_rep[8]    160.45    0.13 5.47  153.41  160.38  167.59  1859 1.00
## y_rep[9]    149.50    0.13 5.58  142.44  149.38  156.55  1946 1.00
## y_rep[10]   160.56    0.12 5.58  153.33  160.39  167.74  2016 1.00
## y_rep[11]   149.64    0.12 5.50  142.73  149.40  156.62  2020 1.00
## y_rep[12]   160.37    0.12 5.42  153.31  160.43  167.33  1950 1.00
## y_rep[13]   149.38    0.13 5.63  142.11  149.30  156.66  1944 1.00
## y_rep[14]   149.43    0.13 5.60  142.16  149.42  156.53  1753 1.00
## y_rep[15]   149.54    0.13 5.51  142.34  149.62  156.64  1797 1.00
## y_rep[16]   160.43    0.12 5.58  153.32  160.41  167.63  2003 1.00
## y_rep[17]   160.50    0.13 5.54  153.58  160.46  167.56  1961 1.00
## y_rep[18]   149.42    0.13 5.53  142.35  149.32  156.68  1797 1.00
## y_rep[19]   160.38    0.13 5.50  153.33  160.48  167.20  1929 1.00
## y_rep[20]   149.51    0.12 5.56  142.48  149.45  156.65  2050 1.00
## y_rep[21]   160.24    0.13 5.71  153.11  160.14  167.85  2025 1.00
## y_rep[22]   149.53    0.13 5.71  142.29  149.54  156.64  1925 1.00
## y_rep[23]   149.63    0.13 5.60  142.62  149.56  156.85  1771 1.00
## y_rep[24]   160.26    0.13 5.73  152.90  160.37  167.44  2012 1.00
## y_rep[25]   160.26    0.12 5.39  153.37  160.17  167.32  1962 1.00
## y_rep[26]   149.39    0.13 5.68  142.05  149.47  156.49  2028 1.00
## y_rep[27]   149.67    0.13 5.65  142.41  149.50  156.83  1957 1.00
## y_rep[28]   149.46    0.12 5.59  142.01  149.52  156.43  2074 1.00
## y_rep[29]   149.46    0.12 5.52  142.61  149.55  156.53  1951 1.00
## y_rep[30]   149.53    0.13 5.63  142.31  149.46  156.78  1935 1.00
## y_rep[31]   149.28    0.12 5.50  142.37  149.18  156.29  2026 1.00
## y_rep[32]   160.39    0.13 5.53  153.44  160.17  167.69  1867 1.00
## y_rep[33]   160.20    0.12 5.54  153.00  160.18  167.08  2001 1.00
## y_rep[34]   160.11    0.13 5.56  152.93  160.28  167.33  1799 1.00
## y_rep[35]   160.30    0.12 5.63  152.94  160.17  167.79  2112 1.00
## y_rep[36]   149.42    0.12 5.45  142.51  149.47  156.54  2035 1.00
## y_rep[37]   149.61    0.12 5.50  142.50  149.62  156.40  2044 1.00
## y_rep[38]   160.30    0.13 5.60  153.04  160.39  167.07  1934 1.00
## y_rep[39]   149.48    0.13 5.57  142.38  149.47  156.52  1729 1.00
## y_rep[40]   160.26    0.12 5.50  153.33  160.35  167.22  2111 1.00
## y_rep[41]   160.28    0.12 5.53  153.23  160.33  167.48  2020 1.00
## y_rep[42]   160.23    0.12 5.58  153.04  160.15  167.39  2205 1.00
## y_rep[43]   149.42    0.13 5.73  142.27  149.34  156.79  1841 1.00
## y_rep[44]   149.52    0.12 5.47  142.79  149.42  156.59  1947 1.00
## y_rep[45]   149.57    0.13 5.59  142.28  149.64  156.55  1979 1.00
## y_rep[46]   149.53    0.13 5.56  142.50  149.54  156.66  1837 1.00
## y_rep[47]   160.41    0.13 5.70  152.96  160.44  167.67  1965 1.00
## y_rep[48]   149.35    0.12 5.58  142.15  149.32  156.40  2038 1.00
## y_rep[49]   160.19    0.14 5.57  153.00  160.19  167.32  1522 1.00
## y_rep[50]   149.42    0.13 5.59  142.28  149.36  156.70  1986 1.00
## y_rep[51]   149.58    0.12 5.61  142.33  149.42  156.85  2055 1.00
## y_rep[52]   149.63    0.13 5.48  142.64  149.63  156.79  1904 1.00
## y_rep[53]   160.31    0.12 5.47  153.45  160.32  167.35  1932 1.00
## y_rep[54]   160.45    0.12 5.63  153.39  160.34  167.63  2067 1.00
## y_rep[55]   149.56    0.14 5.57  142.44  149.60  156.68  1525 1.00
## y_rep[56]   160.40    0.12 5.54  153.32  160.40  167.64  1999 1.00
## y_rep[57]   149.53    0.12 5.40  142.86  149.48  156.51  2109 1.00
## y_rep[58]   160.08    0.12 5.66  152.78  160.16  167.29  2105 1.00
## y_rep[59]   149.71    0.13 5.57  142.53  149.75  156.77  1906 1.00
## y_rep[60]   160.50    0.13 5.66  153.39  160.40  167.71  1801 1.00
## y_rep[61]   149.63    0.12 5.51  142.57  149.54  156.65  2075 1.00
## y_rep[62]   149.71    0.12 5.56  142.60  149.64  156.88  2038 1.00
## y_rep[63]   160.30    0.12 5.52  152.95  160.48  167.13  1980 1.00
## y_rep[64]   160.20    0.12 5.47  153.41  160.15  167.03  2240 1.00
## y_rep[65]   149.46    0.13 5.56  142.41  149.55  156.34  1888 1.00
## y_rep[66]   160.44    0.11 5.35  153.61  160.34  167.37  2166 1.00
## y_rep[67]   149.68    0.14 5.57  142.51  149.78  156.78  1632 1.00
## y_rep[68]   149.45    0.13 5.53  142.42  149.50  156.34  1951 1.00
## y_rep[69]   160.41    0.12 5.45  153.66  160.24  167.49  1939 1.00
## y_rep[70]   149.51    0.12 5.47  142.41  149.55  156.57  2134 1.00
## y_rep[71]   160.74    0.12 5.51  153.66  160.73  167.67  1973 1.00
## y_rep[72]   160.32    0.13 5.49  153.22  160.38  167.08  1823 1.00
## y_rep[73]   160.24    0.13 5.62  152.94  160.29  167.38  1880 1.00
## y_rep[74]   149.38    0.12 5.40  142.35  149.52  156.24  1989 1.00
## y_rep[75]   149.62    0.12 5.61  142.46  149.58  156.80  2025 1.00
## y_rep[76]   149.50    0.12 5.53  142.05  149.70  156.54  2073 1.00
## y_rep[77]   149.51    0.12 5.46  142.36  149.52  156.54  2077 1.00
## y_rep[78]   160.49    0.12 5.59  153.56  160.33  167.64  2073 1.00
## y_rep[79]   149.62    0.13 5.68  142.33  149.38  157.10  1951 1.00
## y_rep[80]   149.51    0.12 5.59  142.26  149.54  156.78  2003 1.00
## y_rep[81]   160.42    0.13 5.48  153.42  160.38  167.39  1889 1.00
## y_rep[82]   160.31    0.12 5.46  153.57  160.12  167.47  2041 1.00
## y_rep[83]   149.34    0.13 5.67  142.18  149.37  156.60  1844 1.00
## y_rep[84]   160.53    0.12 5.61  153.26  160.55  167.49  2030 1.00
## y_rep[85]   149.49    0.12 5.56  142.35  149.28  156.57  2050 1.00
## y_rep[86]   160.31    0.12 5.50  153.11  160.42  167.47  2006 1.00
## y_rep[87]   149.74    0.12 5.46  142.86  149.80  156.88  2126 1.00
## y_rep[88]   160.34    0.12 5.59  152.95  160.31  167.36  2006 1.00
## y_rep[89]   149.49    0.12 5.46  142.59  149.42  156.42  1968 1.00
## y_rep[90]   160.14    0.14 5.59  152.96  160.19  167.16  1597 1.00
## y_rep[91]   160.25    0.12 5.65  153.02  160.16  167.53  2082 1.00
## y_rep[92]   160.40    0.13 5.73  153.09  160.47  167.59  1968 1.00
## y_rep[93]   149.37    0.12 5.60  141.98  149.45  156.33  2021 1.00
## y_rep[94]   149.41    0.13 5.54  142.25  149.42  156.43  1872 1.00
## y_rep[95]   160.43    0.12 5.54  153.05  160.56  167.35  1975 1.00
## y_rep[96]   160.39    0.13 5.77  153.29  160.26  167.80  2008 1.00
## y_rep[97]   149.46    0.14 5.54  142.34  149.55  156.60  1678 1.00
## y_rep[98]   160.47    0.12 5.49  153.42  160.36  167.57  1938 1.00
## y_rep[99]   160.22    0.13 5.66  152.96  160.13  167.37  2032 1.00
## y_rep[100]  149.69    0.12 5.54  142.57  149.66  156.75  2045 1.00
## y_rep[101]  160.54    0.14 5.69  153.43  160.56  167.73  1603 1.00
## y_rep[102]  149.40    0.13 5.70  142.02  149.52  156.66  1954 1.00
## y_rep[103]  149.40    0.13 5.71  142.08  149.33  156.69  1945 1.00
## y_rep[104]  160.56    0.14 5.56  153.61  160.48  167.75  1641 1.00
## y_rep[105]  160.50    0.12 5.46  153.32  160.55  167.45  1946 1.00
## y_rep[106]  149.49    0.12 5.49  142.18  149.62  156.49  2053 1.00
## y_rep[107]  149.74    0.13 5.67  142.50  149.50  157.10  1971 1.00
## y_rep[108]  149.50    0.14 5.60  142.49  149.56  156.43  1563 1.01
## y_rep[109]  160.35    0.12 5.49  153.26  160.31  167.31  2125 1.00
## y_rep[110]  149.48    0.12 5.52  142.40  149.44  156.69  2129 1.00
## y_rep[111]  160.21    0.12 5.58  153.14  160.14  167.49  2075 1.00
## y_rep[112]  149.54    0.13 5.54  142.56  149.50  156.69  1735 1.00
## y_rep[113]  149.40    0.13 5.58  142.42  149.43  156.47  1914 1.00
## y_rep[114]  160.23    0.13 5.62  153.26  160.17  167.24  1964 1.00
## y_rep[115]  160.45    0.12 5.35  153.60  160.35  167.53  1981 1.00
## y_rep[116]  149.43    0.12 5.53  142.31  149.35  156.68  1986 1.00
## y_rep[117]  160.42    0.12 5.55  153.30  160.30  167.64  2082 1.00
## y_rep[118]  160.30    0.12 5.53  153.10  160.46  167.19  1977 1.00
## y_rep[119]  149.53    0.14 5.56  142.52  149.44  156.56  1599 1.00
## y_rep[120]  149.47    0.13 5.64  142.10  149.54  156.64  1965 1.00
## y_rep[121]  160.58    0.12 5.55  153.32  160.63  167.86  2045 1.00
## y_rep[122]  149.47    0.13 5.55  142.28  149.52  156.42  1903 1.00
## y_rep[123]  160.38    0.13 5.49  153.29  160.49  167.34  1721 1.00
## y_rep[124]  149.56    0.12 5.66  142.04  149.49  156.59  2059 1.00
## y_rep[125]  149.42    0.14 5.50  142.51  149.33  156.73  1609 1.00
## y_rep[126]  149.47    0.13 5.58  142.25  149.55  156.61  1858 1.00
## y_rep[127]  149.79    0.12 5.54  142.52  149.93  156.67  2079 1.00
## y_rep[128]  160.06    0.13 5.63  153.03  159.95  167.19  1878 1.00
## y_rep[129]  149.45    0.12 5.47  142.65  149.39  156.49  1974 1.00
## y_rep[130]  149.43    0.12 5.48  142.59  149.47  156.66  1955 1.00
## y_rep[131]  160.32    0.12 5.54  153.18  160.39  167.25  1989 1.00
## y_rep[132]  160.52    0.12 5.45  153.41  160.46  167.62  2073 1.00
## y_rep[133]  149.68    0.12 5.40  142.78  149.69  156.68  2052 1.00
## y_rep[134]  160.38    0.12 5.49  153.32  160.54  167.44  1942 1.00
## y_rep[135]  149.56    0.12 5.50  142.54  149.52  156.73  2104 1.00
## y_rep[136]  160.34    0.13 5.71  153.30  160.14  167.89  2006 1.00
## y_rep[137]  160.35    0.13 5.58  153.05  160.49  167.28  1991 1.00
## y_rep[138]  160.57    0.12 5.52  153.48  160.68  167.56  1963 1.00
## y_rep[139]  149.63    0.13 5.52  142.64  149.70  156.50  1810 1.00
## y_rep[140]  160.50    0.12 5.66  153.26  160.53  167.58  2099 1.00
## y_rep[141]  149.63    0.13 5.52  142.50  149.50  156.84  1700 1.00
## y_rep[142]  149.42    0.14 5.72  142.21  149.28  156.81  1719 1.00
## y_rep[143]  149.64    0.13 5.59  142.62  149.57  156.89  1792 1.00
## y_rep[144]  149.55    0.12 5.56  142.50  149.50  156.45  2059 1.00
## y_rep[145]  149.39    0.12 5.60  142.28  149.40  156.54  2043 1.00
## y_rep[146]  160.29    0.13 5.64  152.98  160.23  167.45  1980 1.00
## y_rep[147]  149.71    0.13 5.50  142.65  149.75  156.67  1903 1.00
## y_rep[148]  149.82    0.12 5.57  142.79  149.65  156.87  2289 1.00
## y_rep[149]  160.33    0.12 5.50  153.40  160.14  167.40  2005 1.00
## y_rep[150]  149.40    0.13 5.51  142.32  149.43  156.50  1880 1.00
## y_rep[151]  160.36    0.13 5.62  153.34  160.32  167.50  1907 1.00
## y_rep[152]  149.50    0.13 5.71  142.15  149.57  156.70  1907 1.00
## y_rep[153]  160.44    0.13 5.43  153.45  160.52  167.27  1805 1.00
## y_rep[154]  149.36    0.12 5.39  142.41  149.40  156.28  2018 1.00
## y_rep[155]  160.35    0.12 5.46  153.42  160.36  167.22  2104 1.00
## y_rep[156]  149.63    0.12 5.60  142.44  149.70  156.80  2089 1.00
## y_rep[157]  149.51    0.13 5.51  142.26  149.51  156.47  1871 1.00
## y_rep[158]  160.58    0.13 5.66  153.47  160.56  167.80  1923 1.00
## y_rep[159]  160.49    0.12 5.52  153.46  160.46  167.71  2164 1.00
## y_rep[160]  149.48    0.12 5.50  142.52  149.48  156.50  1963 1.00
## y_rep[161]  160.48    0.13 5.52  153.16  160.60  167.70  1842 1.00
## y_rep[162]  149.46    0.12 5.65  142.30  149.50  156.70  2073 1.00
## y_rep[163]  160.29    0.13 5.49  153.43  160.27  167.31  1681 1.00
## y_rep[164]  160.25    0.13 5.58  153.15  160.20  167.67  1842 1.00
## y_rep[165]  160.33    0.12 5.44  153.47  160.30  167.17  1997 1.00
## y_rep[166]  149.58    0.13 5.39  142.89  149.53  156.50  1851 1.00
## y_rep[167]  160.38    0.12 5.48  153.37  160.46  167.62  2019 1.00
## y_rep[168]  149.41    0.13 5.58  142.37  149.46  156.81  1952 1.00
## y_rep[169]  149.47    0.13 5.56  142.34  149.46  156.37  1889 1.00
## y_rep[170]  149.70    0.12 5.47  142.91  149.56  156.91  2037 1.00
## y_rep[171]  149.40    0.12 5.54  142.29  149.37  156.36  2156 1.00
## y_rep[172]  160.41    0.12 5.63  153.26  160.48  167.40  2072 1.00
## y_rep[173]  149.55    0.13 5.68  142.18  149.65  156.92  1918 1.00
## y_rep[174]  160.46    0.12 5.47  153.39  160.54  167.53  1976 1.00
## y_rep[175]  149.59    0.12 5.58  142.70  149.68  156.66  2062 1.00
## y_rep[176]  160.47    0.13 5.69  153.22  160.47  167.72  2030 1.00
## y_rep[177]  149.60    0.13 5.67  142.45  149.30  157.02  2021 1.00
## y_rep[178]  149.61    0.12 5.59  142.36  149.66  156.43  2029 1.00
## y_rep[179]  160.43    0.13 5.64  153.25  160.43  167.65  1916 1.00
## y_rep[180]  149.48    0.12 5.54  142.39  149.64  156.60  2193 1.00
## y_rep[181]  160.16    0.13 5.60  152.94  160.36  166.98  1938 1.00
## y_rep[182]  149.60    0.12 5.57  142.35  149.55  156.62  2068 1.00
## y_rep[183]  149.40    0.12 5.48  142.31  149.42  156.32  2083 1.00
## y_rep[184]  149.42    0.12 5.54  142.27  149.51  156.31  2023 1.00
## y_rep[185]  160.43    0.13 5.59  153.28  160.29  167.86  1949 1.00
## y_rep[186]  160.29    0.13 5.56  153.33  160.31  167.31  1746 1.00
## y_rep[187]  149.56    0.12 5.60  142.17  149.55  156.60  2068 1.00
## y_rep[188]  160.46    0.12 5.60  153.23  160.45  167.51  2067 1.00
## y_rep[189]  149.39    0.13 5.54  142.09  149.37  156.39  1841 1.00
## y_rep[190]  160.45    0.13 5.64  153.24  160.38  167.76  1931 1.00
## y_rep[191]  149.34    0.13 5.56  142.29  149.22  156.49  1825 1.00
## y_rep[192]  160.37    0.13 5.54  153.12  160.38  167.34  1808 1.00
## y_rep[193]  149.49    0.13 5.49  142.64  149.51  156.59  1908 1.00
## y_rep[194]  160.39    0.13 5.58  153.28  160.27  167.54  1944 1.00
## y_rep[195]  149.48    0.13 5.63  142.35  149.41  156.73  1969 1.00
## y_rep[196]  149.60    0.13 5.47  142.82  149.49  156.65  1785 1.00
## y_rep[197]  149.59    0.12 5.58  142.53  149.63  156.54  2060 1.00
## y_rep[198]  160.31    0.13 5.50  153.27  160.29  167.51  1930 1.00
## y_rep[199]  160.27    0.13 5.70  153.05  160.29  167.41  2074 1.00
## y_rep[200]  149.41    0.12 5.59  142.26  149.39  156.42  2207 1.00
## y_rep[201]  160.61    0.13 5.54  153.44  160.66  167.83  1829 1.00
## y_rep[202]  149.24    0.12 5.61  141.99  149.26  156.39  2140 1.00
## y_rep[203]  160.36    0.12 5.75  153.15  160.29  167.63  2205 1.00
## y_rep[204]  149.32    0.13 5.56  142.18  149.41  156.19  1881 1.00
## y_rep[205]  160.34    0.13 5.55  153.28  160.32  167.49  1949 1.00
## y_rep[206]  149.55    0.14 5.50  142.58  149.49  156.57  1589 1.00
## y_rep[207]  149.53    0.13 5.73  142.28  149.44  156.95  2090 1.00
## y_rep[208]  149.34    0.12 5.50  142.30  149.55  156.41  1941 1.00
## y_rep[209]  149.45    0.13 5.64  142.19  149.43  156.83  1915 1.00
## y_rep[210]  149.43    0.12 5.53  142.43  149.43  156.51  2051 1.00
## y_rep[211]  160.33    0.12 5.51  153.21  160.45  167.48  2034 1.00
## y_rep[212]  160.34    0.13 5.66  152.85  160.48  167.34  1887 1.00
## y_rep[213]  160.24    0.13 5.55  153.16  160.20  167.27  1785 1.00
## y_rep[214]  149.53    0.12 5.41  142.53  149.57  156.70  1933 1.00
## y_rep[215]  160.36    0.12 5.47  153.29  160.49  167.31  1961 1.00
## y_rep[216]  160.34    0.12 5.55  153.24  160.49  167.30  2132 1.00
## y_rep[217]  160.35    0.13 5.62  153.09  160.48  167.53  1790 1.00
## y_rep[218]  149.58    0.13 5.42  142.68  149.54  156.50  1737 1.00
## y_rep[219]  160.45    0.13 5.65  153.04  160.55  167.58  2013 1.00
## y_rep[220]  149.45    0.12 5.38  142.56  149.45  156.46  1933 1.00
## y_rep[221]  149.46    0.12 5.51  142.36  149.43  156.52  2202 1.00
## y_rep[222]  149.65    0.12 5.41  142.58  149.52  156.79  2082 1.00
## y_rep[223]  160.34    0.13 5.77  152.91  160.48  167.83  2036 1.00
## y_rep[224]  149.40    0.13 5.56  142.56  149.35  156.29  1951 1.00
## y_rep[225]  160.40    0.13 5.67  153.30  160.43  167.74  1910 1.00
## y_rep[226]  149.40    0.12 5.57  142.35  149.43  156.24  1995 1.00
## y_rep[227]  149.48    0.13 5.48  142.18  149.48  156.50  1867 1.00
## y_rep[228]  160.43    0.13 5.59  153.24  160.51  167.56  1766 1.00
## y_rep[229]  149.53    0.13 5.56  142.32  149.44  156.65  1883 1.00
## y_rep[230]  160.30    0.13 5.58  153.26  160.29  167.47  1960 1.00
## y_rep[231]  160.33    0.12 5.47  153.46  160.22  167.33  2051 1.00
## y_rep[232]  149.38    0.13 5.69  141.92  149.36  156.70  1929 1.00
## y_rep[233]  149.50    0.13 5.57  142.56  149.45  156.75  1782 1.00
## y_rep[234]  160.29    0.12 5.43  153.39  160.28  167.08  2059 1.00
## y_rep[235]  160.51    0.12 5.54  153.45  160.45  167.52  1980 1.00
## y_rep[236]  149.65    0.12 5.32  142.81  149.55  156.44  1997 1.00
## y_rep[237]  160.18    0.12 5.57  153.10  160.18  167.31  2169 1.00
## y_rep[238]  149.45    0.13 5.60  142.17  149.47  156.42  1960 1.00
## y_rep[239]  149.51    0.13 5.64  142.27  149.57  156.59  1914 1.00
## y_rep[240]  149.65    0.13 5.52  142.59  149.69  156.72  1721 1.00
## y_rep[241]  149.52    0.12 5.46  142.48  149.70  156.36  1957 1.00
## y_rep[242]  160.54    0.12 5.47  153.61  160.70  167.40  1987 1.00
## y_rep[243]  160.35    0.13 5.63  153.03  160.43  167.45  1879 1.00
## y_rep[244]  149.60    0.13 5.64  142.33  149.76  156.69  1890 1.00
## y_rep[245]  160.22    0.12 5.63  153.32  160.25  167.43  2120 1.00
## y_rep[246]  149.67    0.13 5.69  142.50  149.71  156.87  1891 1.00
## y_rep[247]  160.13    0.12 5.42  153.25  160.06  167.12  1980 1.00
## y_rep[248]  149.53    0.13 5.64  142.33  149.67  156.71  1926 1.00
## y_rep[249]  160.39    0.13 5.66  153.19  160.40  167.78  2016 1.00
## y_rep[250]  160.22    0.13 5.67  152.94  160.24  167.58  1889 1.00
## y_rep[251]  160.38    0.13 5.53  153.12  160.36  167.36  1830 1.00
## y_rep[252]  160.22    0.13 5.63  153.03  160.04  167.53  1955 1.00
## y_rep[253]  149.31    0.13 5.51  142.21  149.33  156.34  1893 1.00
## y_rep[254]  149.33    0.13 5.67  142.07  149.33  156.63  1880 1.00
## y_rep[255]  149.37    0.13 5.73  142.15  149.24  156.72  1999 1.00
## y_rep[256]  149.31    0.13 5.58  142.36  149.17  156.61  1928 1.00
## y_rep[257]  160.51    0.12 5.54  153.14  160.56  167.58  2053 1.00
## y_rep[258]  149.54    0.12 5.62  142.32  149.67  156.59  2183 1.00
## y_rep[259]  160.53    0.13 5.57  153.19  160.65  167.68  1986 1.00
## y_rep[260]  160.32    0.13 5.83  152.91  160.37  167.71  2079 1.00
## y_rep[261]  160.44    0.13 5.46  153.54  160.43  167.39  1814 1.00
## y_rep[262]  149.72    0.13 5.55  142.81  149.61  156.89  1942 1.00
## y_rep[263]  149.49    0.13 5.57  142.34  149.47  156.50  1929 1.00
## y_rep[264]  149.25    0.13 5.62  142.23  149.22  156.51  1978 1.00
## y_rep[265]  160.33    0.14 5.70  153.10  160.53  167.75  1763 1.00
## y_rep[266]  149.36    0.12 5.37  142.56  149.29  156.35  1907 1.00
## y_rep[267]  160.40    0.13 5.51  153.36  160.37  167.66  1736 1.00
## y_rep[268]  149.45    0.12 5.43  142.61  149.39  156.42  1982 1.00
## y_rep[269]  149.54    0.12 5.48  142.54  149.71  156.57  1974 1.00
## y_rep[270]  160.45    0.13 5.58  153.21  160.41  167.74  1787 1.00
## y_rep[271]  160.30    0.13 5.68  153.12  160.32  167.78  2023 1.00
## y_rep[272]  149.40    0.13 5.65  142.04  149.52  156.59  1999 1.00
## y_rep[273]  160.39    0.13 5.55  153.11  160.48  167.61  1700 1.00
## y_rep[274]  160.53    0.12 5.56  153.44  160.39  167.61  1983 1.00
## y_rep[275]  149.57    0.12 5.52  142.69  149.53  156.62  2011 1.00
## y_rep[276]  160.51    0.13 5.47  153.49  160.62  167.42  1913 1.00
## y_rep[277]  149.59    0.13 5.60  142.43  149.50  156.75  1895 1.00
## y_rep[278]  149.27    0.12 5.57  142.24  149.27  156.56  2125 1.00
## y_rep[279]  160.50    0.13 5.64  153.29  160.65  167.74  1807 1.00
## y_rep[280]  149.66    0.13 5.63  142.31  149.66  156.78  1885 1.00
## y_rep[281]  149.53    0.13 5.54  142.37  149.59  156.64  1922 1.00
## y_rep[282]  160.33    0.13 5.71  153.02  160.23  167.51  2069 1.00
## y_rep[283]  149.76    0.12 5.54  142.91  149.69  156.65  2032 1.00
## y_rep[284]  160.50    0.13 5.70  153.22  160.60  167.66  2077 1.00
## y_rep[285]  149.39    0.12 5.44  142.56  149.33  156.57  2102 1.00
## y_rep[286]  160.34    0.13 5.69  153.08  160.35  167.77  1977 1.00
## y_rep[287]  160.43    0.12 5.41  153.51  160.49  167.07  2175 1.00
## y_rep[288]  160.41    0.13 5.61  153.02  160.39  167.69  1860 1.00
## y_rep[289]  149.59    0.13 5.61  142.05  149.60  156.82  1997 1.00
## y_rep[290]  160.29    0.12 5.56  153.19  160.34  167.34  2107 1.00
## y_rep[291]  160.36    0.12 5.71  152.98  160.46  167.74  2193 1.00
## y_rep[292]  160.25    0.13 5.73  153.17  160.15  167.55  1980 1.00
## y_rep[293]  160.38    0.13 5.79  153.07  160.40  167.73  1847 1.00
## y_rep[294]  149.60    0.12 5.56  142.40  149.68  156.53  2004 1.00
## y_rep[295]  160.22    0.13 5.55  153.18  160.24  167.29  1777 1.00
## y_rep[296]  149.51    0.12 5.41  142.83  149.50  156.56  1895 1.00
## y_rep[297]  149.61    0.12 5.53  142.87  149.62  156.57  2026 1.00
## y_rep[298]  149.50    0.13 5.60  142.43  149.40  156.57  1987 1.00
## y_rep[299]  160.46    0.12 5.46  153.43  160.40  167.65  2060 1.00
## y_rep[300]  160.40    0.12 5.48  153.44  160.31  167.46  1945 1.00
## y_rep[301]  149.64    0.13 5.53  142.63  149.65  156.56  1917 1.00
## y_rep[302]  160.32    0.13 5.48  153.41  160.34  167.31  1920 1.00
## y_rep[303]  149.53    0.12 5.42  142.66  149.42  156.46  1897 1.00
## y_rep[304]  160.26    0.13 5.61  152.98  160.31  167.30  1992 1.00
## y_rep[305]  160.19    0.12 5.62  153.22  160.15  167.45  2140 1.00
## y_rep[306]  160.25    0.13 5.63  153.00  160.30  167.30  1956 1.00
## y_rep[307]  160.43    0.12 5.48  153.43  160.26  167.49  1930 1.00
## y_rep[308]  149.63    0.12 5.64  142.51  149.65  156.81  2094 1.00
## y_rep[309]  149.48    0.12 5.52  142.64  149.30  156.60  2034 1.00
## y_rep[310]  149.64    0.13 5.59  142.42  149.73  156.55  1980 1.00
## y_rep[311]  149.41    0.13 5.47  142.49  149.52  156.39  1894 1.00
## y_rep[312]  149.43    0.13 5.48  142.36  149.48  156.47  1756 1.00
## y_rep[313]  160.15    0.12 5.53  153.04  160.10  167.22  2098 1.00
## y_rep[314]  149.37    0.14 5.60  142.25  149.38  156.35  1631 1.00
## y_rep[315]  160.59    0.13 5.49  153.43  160.65  167.51  1812 1.00
## y_rep[316]  149.49    0.13 5.53  142.40  149.44  156.42  1890 1.00
## y_rep[317]  160.33    0.13 5.55  153.31  160.53  167.15  1798 1.00
## y_rep[318]  149.56    0.13 5.58  142.34  149.41  156.79  1894 1.00
## y_rep[319]  160.28    0.12 5.38  153.27  160.30  167.08  2120 1.00
## y_rep[320]  149.52    0.13 5.62  142.34  149.55  156.68  1792 1.00
## y_rep[321]  160.24    0.13 5.53  153.16  160.30  167.44  1913 1.00
## y_rep[322]  149.26    0.12 5.58  142.16  149.03  156.38  2094 1.00
## y_rep[323]  160.47    0.12 5.35  153.74  160.55  167.43  2092 1.00
## y_rep[324]  160.33    0.13 5.54  153.13  160.19  167.36  1870 1.00
## y_rep[325]  149.44    0.12 5.38  142.81  149.43  156.31  1900 1.00
## y_rep[326]  149.30    0.12 5.64  142.10  149.31  156.46  2075 1.00
## y_rep[327]  149.25    0.13 5.64  141.99  149.33  156.27  1980 1.00
## y_rep[328]  160.60    0.13 5.39  153.62  160.71  167.70  1828 1.00
## y_rep[329]  160.26    0.13 5.51  153.24  160.40  167.17  1940 1.00
## y_rep[330]  160.49    0.13 5.72  153.28  160.43  167.97  1946 1.00
## y_rep[331]  160.05    0.12 5.52  152.93  160.12  167.04  2037 1.00
## y_rep[332]  160.36    0.12 5.46  153.38  160.29  167.33  1930 1.00
## y_rep[333]  149.58    0.13 5.58  142.50  149.66  156.78  1940 1.00
## y_rep[334]  160.31    0.13 5.45  153.15  160.36  167.02  1837 1.00
## y_rep[335]  149.71    0.12 5.55  142.59  149.71  156.69  2064 1.00
## y_rep[336]  160.31    0.12 5.65  152.95  160.45  167.58  2083 1.00
## y_rep[337]  149.44    0.13 5.66  142.23  149.40  156.85  2005 1.00
## y_rep[338]  149.58    0.12 5.58  142.65  149.36  156.89  2150 1.00
## y_rep[339]  149.65    0.13 5.57  142.58  149.68  156.83  1966 1.00
## y_rep[340]  149.43    0.12 5.34  142.64  149.38  156.28  1978 1.00
## y_rep[341]  149.51    0.12 5.40  142.62  149.54  156.39  2085 1.00
## y_rep[342]  160.29    0.12 5.61  153.07  160.43  167.36  2037 1.00
## y_rep[343]  149.50    0.13 5.66  142.24  149.56  156.57  2045 1.00
## y_rep[344]  160.35    0.12 5.55  153.21  160.38  167.47  2059 1.00
## y_rep[345]  149.41    0.12 5.59  142.31  149.39  156.51  2109 1.00
## y_rep[346]  160.61    0.12 5.41  153.79  160.62  167.55  2099 1.00
## y_rep[347]  149.42    0.13 5.51  142.24  149.33  156.61  1790 1.00
## y_rep[348]  149.49    0.12 5.37  142.72  149.44  156.48  2134 1.00
## y_rep[349]  149.56    0.13 5.59  142.52  149.45  156.62  1980 1.00
## y_rep[350]  160.36    0.12 5.44  153.34  160.35  167.32  2011 1.00
## y_rep[351]  149.45    0.12 5.48  142.33  149.47  156.47  1961 1.00
## y_rep[352]  160.26    0.12 5.48  153.18  160.33  167.34  1976 1.00
## lp__       -779.77    0.04 1.27 -781.47 -779.41 -778.55  1051 1.00
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 15:25:46 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Corresponds to a frequentist model with the intercept fixed to zero:

summary(lm(height~as.factor(male)-1,data=howell2))
## 
## Call:
## lm(formula = height ~ as.factor(male) - 1, data = howell2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.6585  -3.4635   0.2965   3.4715  18.7115 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## as.factor(male)0 149.5135     0.4049   369.2   <2e-16 ***
## as.factor(male)1 160.3585     0.4311   372.0   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.537 on 350 degrees of freedom
## Multiple R-squared:  0.9987, Adjusted R-squared:  0.9987 
## F-statistic: 1.374e+05 on 2 and 350 DF,  p-value: < 2.2e-16

Even if you use a index variable (with values 1 or 2), you can also use the same notation as for dummy variables

X <- model.matrix(~male.index, howell2)
m5.2.bis.data <- list(N = nrow(howell2), height = howell2$height, X = X, n = length(unique(howell2$male.index)))
str(m5.2.bis.data)
## List of 4
##  $ N     : int 352
##  $ height: num [1:352] 152 140 137 157 145 ...
##  $ X     : num [1:352, 1:2] 1 1 1 1 1 1 1 1 1 1 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:352] "1" "2" "3" "4" ...
##   .. ..$ : chr [1:2] "(Intercept)" "male.index"
##   ..- attr(*, "assign")= int [1:2] 0 1
##  $ n     : int 2
data {                                                     
  int<lower=0> N;                                          // number of observations
  int<lower=0> n;                                          // number of levels of the categorical variable
  vector[N] height;                                        // response variables
  matrix[N,n] X;                                           // categorical variable
}

parameters {                                          
  vector[n] beta;                                         // intercepts of the categorical variable
  real sigma;
}

transformed parameters {
vector[N] mu;                                             
mu = X*beta;  
}

model {
  height ~ normal(mu ,sigma);                              // likelihood
  sigma ~ cauchy(0,5);
  beta[1] ~ normal(178,20);
  beta[2] ~ normal(0,10);
}

generated quantities {
  vector[N] y_rep;
 for(i in 1:N)
 y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_2_bis <- sampling(m5.2.bis, data = m5.2.bis.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_2_bis, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: ea1df6c828dbad449e9a4c6c6321058c.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     138.78    0.04 0.92  137.63  138.78  139.97   641 1.00
## beta[2]      10.77    0.02 0.59   10.00   10.76   11.53   623 1.00
## sigma         5.55    0.01 0.21    5.29    5.54    5.82  1130 1.00
## mu[1]       160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[2]       149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[3]       149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[4]       160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[5]       149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[6]       160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[7]       149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[8]       160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[9]       149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[10]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[11]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[12]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[13]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[14]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[15]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[16]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[17]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[18]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[19]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[20]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[21]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[22]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[23]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[24]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[25]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[26]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[27]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[28]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[29]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[30]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[31]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[32]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[33]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[34]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[35]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[36]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[37]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[38]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[39]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[40]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[41]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[42]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[43]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[44]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[45]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[46]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[47]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[48]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[49]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[50]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[51]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[52]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[53]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[54]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[55]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[56]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[57]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[58]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[59]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[60]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[61]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[62]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[63]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[64]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[65]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[66]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[67]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[68]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[69]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[70]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[71]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[72]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[73]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[74]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[75]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[76]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[77]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[78]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[79]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[80]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[81]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[82]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[83]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[84]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[85]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[86]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[87]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[88]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[89]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[90]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[91]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[92]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[93]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[94]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[95]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[96]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[97]      149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[98]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[99]      160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[100]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[101]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[102]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[103]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[104]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[105]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[106]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[107]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[108]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[109]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[110]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[111]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[112]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[113]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[114]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[115]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[116]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[117]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[118]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[119]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[120]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[121]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[122]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[123]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[124]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[125]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[126]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[127]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[128]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[129]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[130]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[131]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[132]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[133]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[134]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[135]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[136]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[137]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[138]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[139]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[140]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[141]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[142]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[143]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[144]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[145]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[146]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[147]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[148]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[149]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[150]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[151]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[152]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[153]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[154]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[155]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[156]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[157]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[158]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[159]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[160]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[161]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[162]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[163]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[164]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[165]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[166]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[167]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[168]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[169]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[170]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[171]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[172]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[173]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[174]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[175]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[176]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[177]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[178]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[179]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[180]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[181]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[182]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[183]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[184]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[185]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[186]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[187]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[188]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[189]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[190]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[191]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[192]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[193]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[194]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[195]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[196]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[197]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[198]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[199]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[200]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[201]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[202]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[203]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[204]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[205]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[206]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[207]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[208]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[209]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[210]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[211]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[212]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[213]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[214]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[215]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[216]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[217]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[218]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[219]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[220]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[221]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[222]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[223]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[224]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[225]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[226]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[227]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[228]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[229]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[230]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[231]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[232]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[233]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[234]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[235]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[236]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[237]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[238]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[239]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[240]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[241]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[242]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[243]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[244]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[245]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[246]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[247]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[248]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[249]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[250]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[251]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[252]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[253]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[254]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[255]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[256]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[257]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[258]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[259]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[260]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[261]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[262]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[263]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[264]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[265]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[266]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[267]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[268]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[269]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[270]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[271]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[272]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[273]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[274]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[275]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[276]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[277]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[278]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[279]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[280]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[281]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[282]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[283]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[284]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[285]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[286]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[287]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[288]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[289]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[290]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[291]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[292]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[293]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[294]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[295]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[296]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[297]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[298]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[299]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[300]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[301]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[302]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[303]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[304]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[305]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[306]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[307]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[308]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[309]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[310]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[311]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[312]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[313]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[314]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[315]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[316]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[317]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[318]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[319]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[320]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[321]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[322]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[323]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[324]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[325]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[326]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[327]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[328]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[329]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[330]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[331]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[332]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[333]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[334]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[335]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[336]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[337]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[338]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[339]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[340]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[341]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[342]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[343]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[344]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[345]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[346]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[347]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[348]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[349]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[350]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## mu[351]     149.55    0.01 0.40  149.04  149.54  150.05   981 1.00
## mu[352]     160.31    0.01 0.43  159.76  160.30  160.87  1246 1.00
## y_rep[1]    160.48    0.13 5.49  153.49  160.57  167.45  1821 1.00
## y_rep[2]    149.67    0.12 5.55  142.50  149.66  156.78  2059 1.00
## y_rep[3]    149.44    0.14 5.67  142.21  149.39  156.77  1749 1.00
## y_rep[4]    160.14    0.14 5.46  153.37  160.24  167.06  1540 1.00
## y_rep[5]    149.58    0.13 5.64  142.35  149.60  156.98  1889 1.00
## y_rep[6]    160.20    0.12 5.52  153.16  160.26  167.22  2145 1.00
## y_rep[7]    149.73    0.12 5.58  142.35  149.78  156.84  2133 1.00
## y_rep[8]    160.35    0.13 5.66  153.30  160.33  167.38  1834 1.00
## y_rep[9]    149.40    0.14 5.35  142.48  149.45  156.17  1443 1.00
## y_rep[10]   160.43    0.12 5.57  153.13  160.54  167.65  2145 1.00
## y_rep[11]   149.48    0.13 5.65  142.52  149.36  156.65  1837 1.00
## y_rep[12]   160.12    0.13 5.53  152.65  160.15  167.04  1841 1.00
## y_rep[13]   149.26    0.12 5.47  142.22  149.33  156.20  1964 1.00
## y_rep[14]   149.56    0.12 5.56  142.68  149.56  156.64  2001 1.00
## y_rep[15]   149.58    0.13 5.69  142.39  149.49  157.24  1797 1.00
## y_rep[16]   160.33    0.13 5.74  153.00  160.37  167.62  2001 1.00
## y_rep[17]   160.45    0.13 5.51  153.34  160.31  167.63  1938 1.00
## y_rep[18]   149.58    0.14 5.61  142.54  149.59  156.54  1704 1.00
## y_rep[19]   160.34    0.13 5.57  153.16  160.19  167.74  1963 1.00
## y_rep[20]   149.63    0.12 5.57  142.72  149.58  156.59  2162 1.00
## y_rep[21]   160.54    0.13 5.65  153.21  160.41  167.89  1998 1.00
## y_rep[22]   149.59    0.13 5.66  142.20  149.57  156.88  1811 1.00
## y_rep[23]   149.58    0.13 5.52  142.56  149.54  156.79  1876 1.00
## y_rep[24]   160.28    0.12 5.39  153.42  160.26  167.17  1865 1.00
## y_rep[25]   160.30    0.13 5.57  153.08  160.28  167.20  1962 1.00
## y_rep[26]   149.81    0.13 5.68  142.56  149.76  157.09  1802 1.00
## y_rep[27]   149.76    0.13 5.70  142.57  149.81  156.86  1967 1.00
## y_rep[28]   149.64    0.12 5.61  142.23  149.81  156.60  2145 1.00
## y_rep[29]   149.35    0.12 5.57  142.15  149.32  156.47  2015 1.00
## y_rep[30]   149.64    0.13 5.66  142.35  149.75  156.84  2026 1.00
## y_rep[31]   149.61    0.13 5.58  142.24  149.72  156.77  1879 1.00
## y_rep[32]   160.42    0.13 5.74  153.03  160.40  167.63  1957 1.00
## y_rep[33]   160.26    0.13 5.60  153.06  160.25  167.28  1913 1.00
## y_rep[34]   160.28    0.13 5.57  153.10  160.29  167.17  1905 1.00
## y_rep[35]   160.34    0.12 5.49  153.44  160.16  167.34  2009 1.00
## y_rep[36]   149.44    0.13 5.52  142.27  149.45  156.51  1697 1.00
## y_rep[37]   149.57    0.13 5.46  142.33  149.47  156.59  1802 1.00
## y_rep[38]   160.43    0.13 5.69  153.26  160.33  167.76  1785 1.00
## y_rep[39]   149.60    0.12 5.55  142.22  149.58  156.80  1993 1.00
## y_rep[40]   160.17    0.13 5.54  153.07  160.29  167.38  1955 1.00
## y_rep[41]   160.21    0.12 5.59  153.02  160.31  167.34  2071 1.00
## y_rep[42]   160.29    0.13 5.61  153.18  160.18  167.59  1980 1.00
## y_rep[43]   149.56    0.13 5.60  142.35  149.63  156.61  1845 1.00
## y_rep[44]   149.57    0.13 5.56  142.35  149.57  156.45  1920 1.00
## y_rep[45]   149.42    0.13 5.73  141.89  149.55  156.50  1901 1.00
## y_rep[46]   149.75    0.12 5.57  142.64  149.72  157.08  1999 1.00
## y_rep[47]   160.29    0.13 5.74  153.08  160.28  167.54  1847 1.00
## y_rep[48]   149.41    0.12 5.45  142.47  149.28  156.55  2005 1.00
## y_rep[49]   160.38    0.12 5.48  153.32  160.48  167.26  1958 1.00
## y_rep[50]   149.79    0.13 5.58  142.86  149.73  156.95  1717 1.00
## y_rep[51]   149.42    0.14 5.46  142.52  149.60  156.28  1605 1.00
## y_rep[52]   149.69    0.13 5.58  142.36  149.77  156.65  1904 1.00
## y_rep[53]   160.33    0.13 5.52  153.07  160.36  167.39  1678 1.00
## y_rep[54]   160.42    0.12 5.49  153.27  160.26  167.38  2138 1.00
## y_rep[55]   149.49    0.13 5.69  142.20  149.41  156.61  1853 1.00
## y_rep[56]   160.12    0.13 5.46  153.26  160.03  167.29  1860 1.00
## y_rep[57]   149.41    0.12 5.60  142.35  149.41  156.90  2099 1.00
## y_rep[58]   160.22    0.13 5.58  153.07  160.21  167.38  1921 1.00
## y_rep[59]   149.21    0.13 5.69  141.82  149.32  156.45  1999 1.00
## y_rep[60]   160.04    0.14 5.71  152.77  160.13  167.49  1777 1.00
## y_rep[61]   149.59    0.13 5.58  142.33  149.57  156.83  1920 1.00
## y_rep[62]   149.43    0.13 5.66  142.30  149.39  156.76  1928 1.00
## y_rep[63]   160.09    0.14 5.76  152.80  159.94  167.59  1727 1.00
## y_rep[64]   160.30    0.13 5.67  152.97  160.40  167.38  2010 1.00
## y_rep[65]   149.59    0.12 5.55  142.47  149.55  156.78  2006 1.00
## y_rep[66]   160.05    0.12 5.37  153.11  160.04  166.97  2033 1.00
## y_rep[67]   149.45    0.12 5.54  142.45  149.45  156.57  1963 1.00
## y_rep[68]   149.47    0.14 5.45  142.52  149.69  156.44  1557 1.00
## y_rep[69]   160.51    0.12 5.64  153.10  160.66  167.79  2051 1.00
## y_rep[70]   149.50    0.12 5.62  142.18  149.46  156.82  2083 1.00
## y_rep[71]   160.32    0.12 5.51  153.17  160.44  167.28  1982 1.00
## y_rep[72]   160.02    0.13 5.76  152.40  160.00  167.45  1920 1.00
## y_rep[73]   160.18    0.12 5.62  152.83  160.14  167.33  2121 1.00
## y_rep[74]   149.56    0.13 5.68  142.16  149.61  156.71  1853 1.00
## y_rep[75]   149.54    0.12 5.48  142.46  149.47  156.57  2085 1.00
## y_rep[76]   149.51    0.14 5.58  142.24  149.45  156.70  1688 1.00
## y_rep[77]   149.58    0.12 5.62  142.53  149.49  156.83  2121 1.00
## y_rep[78]   160.21    0.13 5.58  152.98  160.17  167.52  1952 1.00
## y_rep[79]   149.73    0.13 5.69  142.68  149.71  156.86  1925 1.00
## y_rep[80]   149.44    0.13 5.61  142.48  149.37  156.58  1950 1.00
## y_rep[81]   160.06    0.13 5.68  152.87  159.94  167.24  1920 1.00
## y_rep[82]   160.21    0.12 5.52  153.20  160.18  167.29  2130 1.00
## y_rep[83]   149.54    0.12 5.49  142.27  149.54  156.37  1977 1.00
## y_rep[84]   160.46    0.12 5.64  153.43  160.41  167.66  2132 1.00
## y_rep[85]   149.32    0.12 5.47  142.39  149.34  156.26  2014 1.00
## y_rep[86]   160.23    0.12 5.54  153.21  160.12  167.35  2071 1.00
## y_rep[87]   149.55    0.14 5.51  142.62  149.73  156.51  1621 1.00
## y_rep[88]   160.59    0.12 5.50  153.53  160.56  167.64  1977 1.00
## y_rep[89]   149.57    0.12 5.52  142.63  149.53  156.71  2114 1.00
## y_rep[90]   160.30    0.13 5.79  152.75  160.24  167.55  2111 1.00
## y_rep[91]   160.19    0.12 5.39  153.34  159.93  167.07  1997 1.00
## y_rep[92]   160.26    0.13 5.62  153.08  160.38  167.26  1972 1.00
## y_rep[93]   149.59    0.13 5.40  142.67  149.69  156.59  1614 1.00
## y_rep[94]   149.49    0.13 5.52  142.49  149.62  156.33  1922 1.00
## y_rep[95]   160.39    0.12 5.50  153.29  160.34  167.43  1967 1.00
## y_rep[96]   160.49    0.13 5.48  153.13  160.49  167.59  1865 1.00
## y_rep[97]   149.54    0.13 5.65  142.18  149.70  156.56  2041 1.00
## y_rep[98]   160.19    0.12 5.58  152.96  160.28  167.33  2128 1.00
## y_rep[99]   160.48    0.14 5.63  153.31  160.45  167.91  1737 1.00
## y_rep[100]  149.42    0.13 5.45  142.31  149.57  156.28  1671 1.00
## y_rep[101]  160.40    0.13 5.62  153.34  160.52  167.40  1988 1.00
## y_rep[102]  149.52    0.13 5.58  142.30  149.71  156.39  1987 1.00
## y_rep[103]  149.39    0.12 5.58  142.19  149.56  156.52  2032 1.00
## y_rep[104]  160.27    0.12 5.60  153.23  160.17  167.54  2080 1.00
## y_rep[105]  160.33    0.13 5.61  153.32  160.19  167.50  2005 1.00
## y_rep[106]  149.56    0.13 5.62  142.36  149.71  156.64  1982 1.00
## y_rep[107]  149.53    0.12 5.54  142.52  149.58  156.82  2150 1.00
## y_rep[108]  149.61    0.12 5.48  142.58  149.58  156.81  2051 1.00
## y_rep[109]  160.10    0.12 5.63  153.01  160.05  167.27  2100 1.00
## y_rep[110]  149.62    0.12 5.51  142.48  149.61  156.60  1992 1.00
## y_rep[111]  160.28    0.13 5.61  153.19  160.39  167.34  1897 1.00
## y_rep[112]  149.58    0.12 5.49  142.42  149.70  156.53  2047 1.00
## y_rep[113]  149.53    0.13 5.58  142.28  149.60  156.63  1911 1.00
## y_rep[114]  160.10    0.13 5.45  153.23  160.06  166.99  1812 1.00
## y_rep[115]  160.23    0.13 5.60  153.06  160.26  167.48  1820 1.00
## y_rep[116]  149.66    0.13 5.56  142.46  149.82  156.65  1887 1.00
## y_rep[117]  160.35    0.13 5.60  153.12  160.37  167.61  1855 1.00
## y_rep[118]  160.19    0.13 5.70  152.94  160.27  167.36  1964 1.00
## y_rep[119]  149.47    0.12 5.58  142.32  149.33  156.63  2037 1.00
## y_rep[120]  149.74    0.13 5.64  142.39  149.71  156.89  1959 1.00
## y_rep[121]  160.29    0.13 5.52  153.22  160.30  167.39  1901 1.00
## y_rep[122]  149.59    0.13 5.60  142.30  149.61  156.82  1949 1.00
## y_rep[123]  160.24    0.12 5.59  153.10  160.28  167.09  2013 1.00
## y_rep[124]  149.52    0.13 5.59  142.68  149.32  156.72  1965 1.00
## y_rep[125]  149.52    0.12 5.51  142.52  149.46  156.56  2035 1.00
## y_rep[126]  149.58    0.13 5.62  142.40  149.50  156.79  1998 1.00
## y_rep[127]  149.59    0.12 5.56  142.68  149.53  156.49  2256 1.00
## y_rep[128]  160.41    0.13 5.62  153.22  160.42  167.59  1781 1.00
## y_rep[129]  149.38    0.12 5.46  142.15  149.50  156.35  1944 1.00
## y_rep[130]  149.52    0.13 5.57  142.38  149.40  156.67  1916 1.00
## y_rep[131]  160.23    0.13 5.73  152.94  160.11  167.61  1844 1.00
## y_rep[132]  160.24    0.12 5.55  152.99  160.41  167.07  2001 1.00
## y_rep[133]  149.63    0.12 5.50  142.81  149.66  156.48  1952 1.00
## y_rep[134]  160.44    0.12 5.63  153.16  160.44  167.46  2033 1.00
## y_rep[135]  149.52    0.13 5.55  142.27  149.55  156.46  1897 1.00
## y_rep[136]  160.06    0.13 5.70  152.72  159.92  167.58  1878 1.00
## y_rep[137]  160.36    0.13 5.67  153.15  160.53  167.42  1889 1.00
## y_rep[138]  160.45    0.12 5.50  153.46  160.36  167.63  2001 1.00
## y_rep[139]  149.66    0.13 5.68  142.13  149.62  156.97  1897 1.00
## y_rep[140]  160.26    0.12 5.53  153.29  160.20  167.36  2125 1.00
## y_rep[141]  149.59    0.12 5.67  142.49  149.62  156.78  2109 1.00
## y_rep[142]  149.41    0.12 5.48  142.30  149.52  156.35  2029 1.00
## y_rep[143]  149.46    0.12 5.54  142.74  149.41  156.44  1982 1.00
## y_rep[144]  149.57    0.13 5.57  142.29  149.70  156.65  1896 1.00
## y_rep[145]  149.55    0.12 5.58  142.33  149.65  156.42  2038 1.00
## y_rep[146]  160.39    0.13 5.41  153.44  160.32  167.30  1826 1.00
## y_rep[147]  149.44    0.13 5.67  142.30  149.29  156.74  2012 1.00
## y_rep[148]  149.61    0.13 5.67  142.20  149.60  156.89  2002 1.00
## y_rep[149]  160.46    0.12 5.63  153.47  160.38  167.56  2208 1.00
## y_rep[150]  149.42    0.13 5.60  142.03  149.44  156.57  1934 1.00
## y_rep[151]  160.07    0.13 5.59  153.10  159.95  167.04  1949 1.00
## y_rep[152]  149.71    0.13 5.56  142.52  149.75  156.92  1898 1.00
## y_rep[153]  160.28    0.13 5.48  153.23  160.30  167.26  1650 1.00
## y_rep[154]  149.57    0.13 5.54  142.37  149.58  156.70  1743 1.00
## y_rep[155]  160.48    0.12 5.50  153.69  160.49  167.57  2112 1.00
## y_rep[156]  149.60    0.13 5.52  142.44  149.68  156.71  1800 1.00
## y_rep[157]  149.63    0.13 5.53  142.57  149.55  156.60  1936 1.00
## y_rep[158]  160.49    0.13 5.50  153.60  160.45  167.46  1804 1.00
## y_rep[159]  160.38    0.13 5.52  153.28  160.30  167.52  1800 1.00
## y_rep[160]  149.62    0.12 5.50  142.57  149.65  156.75  2115 1.00
## y_rep[161]  160.50    0.12 5.52  153.26  160.46  167.59  1973 1.00
## y_rep[162]  149.31    0.13 5.72  141.74  149.38  156.60  1930 1.00
## y_rep[163]  160.29    0.13 5.72  152.86  160.44  167.50  1983 1.00
## y_rep[164]  160.10    0.13 5.74  152.82  159.99  167.35  1948 1.00
## y_rep[165]  160.29    0.13 5.54  153.13  160.41  167.21  1965 1.00
## y_rep[166]  149.75    0.13 5.62  142.38  149.73  156.98  1995 1.00
## y_rep[167]  160.23    0.13 5.55  153.38  160.23  167.51  1965 1.00
## y_rep[168]  149.58    0.12 5.59  142.29  149.61  156.79  2110 1.00
## y_rep[169]  149.46    0.13 5.52  142.66  149.44  156.51  1936 1.00
## y_rep[170]  149.39    0.14 5.59  142.23  149.38  156.66  1665 1.00
## y_rep[171]  149.39    0.13 5.61  142.19  149.44  156.51  1820 1.00
## y_rep[172]  160.35    0.12 5.50  153.45  160.37  167.50  2023 1.00
## y_rep[173]  149.72    0.13 5.58  142.41  149.89  156.59  1872 1.00
## y_rep[174]  160.39    0.12 5.55  153.24  160.44  167.28  1971 1.00
## y_rep[175]  149.50    0.13 5.73  142.22  149.52  156.95  1940 1.00
## y_rep[176]  160.22    0.12 5.60  152.97  160.39  167.40  2059 1.00
## y_rep[177]  149.41    0.13 5.55  142.45  149.18  156.69  1914 1.00
## y_rep[178]  149.61    0.13 5.67  142.21  149.63  156.81  1968 1.00
## y_rep[179]  160.49    0.13 5.79  152.97  160.54  168.03  2056 1.00
## y_rep[180]  149.49    0.12 5.45  142.53  149.44  156.34  2060 1.00
## y_rep[181]  160.20    0.13 5.51  153.11  160.03  167.09  1877 1.00
## y_rep[182]  149.35    0.12 5.73  142.04  149.31  156.72  2138 1.00
## y_rep[183]  149.35    0.13 5.43  142.39  149.33  156.05  1844 1.00
## y_rep[184]  149.48    0.13 5.67  142.10  149.65  156.83  1998 1.00
## y_rep[185]  160.38    0.12 5.48  153.26  160.30  167.48  2064 1.00
## y_rep[186]  160.37    0.12 5.55  153.19  160.32  167.57  1975 1.00
## y_rep[187]  149.65    0.13 5.57  142.66  149.61  156.88  1721 1.00
## y_rep[188]  160.41    0.12 5.46  153.52  160.34  167.46  1985 1.00
## y_rep[189]  149.75    0.13 5.67  142.46  149.80  157.00  1918 1.00
## y_rep[190]  160.14    0.13 5.55  152.94  160.15  167.13  1939 1.00
## y_rep[191]  149.65    0.13 5.52  142.54  149.65  156.76  1869 1.00
## y_rep[192]  160.35    0.13 5.60  153.06  160.28  167.44  1946 1.00
## y_rep[193]  149.46    0.12 5.48  142.31  149.53  156.34  2012 1.00
## y_rep[194]  160.13    0.13 5.53  152.95  160.09  167.32  1954 1.00
## y_rep[195]  149.56    0.13 5.61  142.45  149.47  156.98  1968 1.00
## y_rep[196]  149.19    0.13 5.60  142.23  149.28  156.51  1785 1.00
## y_rep[197]  149.59    0.14 5.56  142.44  149.70  156.59  1567 1.00
## y_rep[198]  160.43    0.12 5.50  153.37  160.34  167.75  2052 1.00
## y_rep[199]  160.27    0.12 5.55  153.20  160.25  167.44  1981 1.00
## y_rep[200]  149.37    0.13 5.60  142.31  149.22  156.57  1935 1.00
## y_rep[201]  160.39    0.13 5.65  153.27  160.34  167.42  1820 1.00
## y_rep[202]  149.63    0.13 5.60  142.45  149.67  156.51  1908 1.00
## y_rep[203]  160.29    0.13 5.54  153.13  160.25  167.57  1895 1.00
## y_rep[204]  149.57    0.13 5.71  142.34  149.55  156.89  1873 1.00
## y_rep[205]  160.31    0.13 5.76  152.99  160.24  167.88  1917 1.00
## y_rep[206]  149.62    0.12 5.53  142.71  149.55  156.62  2050 1.00
## y_rep[207]  149.43    0.13 5.42  142.61  149.38  156.53  1829 1.00
## y_rep[208]  149.40    0.13 5.61  142.27  149.43  156.68  1951 1.00
## y_rep[209]  149.58    0.12 5.55  142.33  149.70  156.58  2002 1.00
## y_rep[210]  149.50    0.12 5.56  142.42  149.42  156.77  2075 1.00
## y_rep[211]  160.42    0.12 5.59  153.48  160.36  167.37  2049 1.00
## y_rep[212]  160.20    0.12 5.44  152.99  160.33  166.99  1931 1.00
## y_rep[213]  160.15    0.13 5.51  153.44  159.88  167.00  1755 1.00
## y_rep[214]  149.62    0.13 5.63  142.05  149.84  156.66  1795 1.00
## y_rep[215]  160.35    0.12 5.52  153.26  160.39  167.26  2116 1.00
## y_rep[216]  160.51    0.13 5.54  153.34  160.54  167.60  1831 1.00
## y_rep[217]  160.11    0.12 5.45  153.14  160.14  167.03  2136 1.00
## y_rep[218]  149.69    0.12 5.51  142.65  149.70  156.74  2137 1.00
## y_rep[219]  160.04    0.14 5.57  152.87  160.08  167.16  1621 1.00
## y_rep[220]  149.60    0.13 5.62  142.42  149.63  156.79  1991 1.00
## y_rep[221]  149.61    0.12 5.40  142.75  149.55  156.57  1954 1.00
## y_rep[222]  149.60    0.12 5.54  142.60  149.56  156.74  2039 1.00
## y_rep[223]  160.35    0.12 5.47  153.47  160.40  167.12  1955 1.00
## y_rep[224]  149.66    0.12 5.62  142.59  149.69  156.81  2046 1.00
## y_rep[225]  160.51    0.12 5.53  153.66  160.35  167.62  2051 1.00
## y_rep[226]  149.55    0.11 5.48  142.59  149.51  156.53  2401 1.00
## y_rep[227]  149.29    0.12 5.62  142.39  149.18  156.39  2062 1.00
## y_rep[228]  160.36    0.13 5.53  153.47  160.37  167.28  1756 1.00
## y_rep[229]  149.50    0.13 5.60  142.30  149.53  156.61  1943 1.00
## y_rep[230]  160.29    0.13 5.58  153.11  160.32  167.61  1874 1.00
## y_rep[231]  160.22    0.13 5.64  153.27  160.21  167.57  1759 1.00
## y_rep[232]  149.57    0.13 5.65  142.34  149.42  156.81  1794 1.00
## y_rep[233]  149.29    0.13 5.57  141.96  149.44  156.45  1970 1.00
## y_rep[234]  160.21    0.13 5.60  153.08  160.13  167.38  1983 1.00
## y_rep[235]  160.07    0.13 5.52  153.05  160.13  167.02  1936 1.00
## y_rep[236]  149.75    0.13 5.44  142.49  149.66  156.82  1883 1.00
## y_rep[237]  160.37    0.12 5.41  153.44  160.29  167.34  2059 1.00
## y_rep[238]  149.59    0.13 5.53  142.67  149.65  156.68  1797 1.00
## y_rep[239]  149.53    0.12 5.59  142.44  149.50  156.95  2083 1.00
## y_rep[240]  149.38    0.12 5.50  142.46  149.38  156.44  1986 1.00
## y_rep[241]  149.52    0.12 5.55  142.48  149.46  156.57  1996 1.00
## y_rep[242]  160.39    0.12 5.63  153.00  160.47  167.44  2086 1.00
## y_rep[243]  160.65    0.13 5.47  153.74  160.72  167.81  1825 1.00
## y_rep[244]  149.48    0.13 5.66  142.36  149.39  156.82  1981 1.00
## y_rep[245]  160.29    0.12 5.71  152.83  160.31  167.54  2226 1.00
## y_rep[246]  149.50    0.13 5.73  142.28  149.62  156.87  1889 1.00
## y_rep[247]  160.20    0.12 5.56  153.27  160.15  167.09  2023 1.00
## y_rep[248]  149.67    0.13 5.62  142.20  149.71  156.85  1819 1.00
## y_rep[249]  160.29    0.13 5.48  153.24  160.22  167.35  1874 1.00
## y_rep[250]  160.33    0.13 5.69  152.98  160.38  167.76  2013 1.00
## y_rep[251]  160.25    0.12 5.51  153.25  160.30  167.45  2074 1.00
## y_rep[252]  160.25    0.12 5.58  153.15  160.27  167.10  2182 1.00
## y_rep[253]  149.35    0.13 5.68  141.96  149.33  156.64  1981 1.00
## y_rep[254]  149.65    0.13 5.47  142.70  149.76  156.52  1899 1.00
## y_rep[255]  149.63    0.12 5.53  142.70  149.52  156.75  2022 1.00
## y_rep[256]  149.53    0.13 5.66  142.19  149.52  156.91  1918 1.00
## y_rep[257]  160.41    0.13 5.52  153.23  160.47  167.37  1948 1.00
## y_rep[258]  149.40    0.12 5.57  142.36  149.33  156.57  2018 1.00
## y_rep[259]  160.06    0.13 5.69  152.91  160.06  167.14  1948 1.00
## y_rep[260]  160.35    0.12 5.49  153.37  160.31  167.56  1954 1.00
## y_rep[261]  160.34    0.12 5.55  153.05  160.33  167.48  2162 1.00
## y_rep[262]  149.60    0.12 5.36  142.83  149.58  156.37  1940 1.00
## y_rep[263]  149.55    0.13 5.55  142.73  149.56  156.52  1889 1.00
## y_rep[264]  149.63    0.13 5.55  142.75  149.54  156.95  1846 1.00
## y_rep[265]  160.35    0.13 5.50  153.40  160.43  167.26  1811 1.00
## y_rep[266]  149.38    0.13 5.57  142.42  149.17  156.47  1826 1.00
## y_rep[267]  160.28    0.13 5.64  153.18  160.29  167.61  1846 1.00
## y_rep[268]  149.60    0.14 5.65  142.64  149.52  156.72  1557 1.00
## y_rep[269]  149.53    0.13 5.62  142.43  149.45  156.73  1847 1.00
## y_rep[270]  160.41    0.13 5.47  153.51  160.49  167.31  1675 1.00
## y_rep[271]  160.33    0.13 5.64  153.12  160.38  167.42  1995 1.00
## y_rep[272]  149.44    0.12 5.51  142.40  149.52  156.47  2178 1.00
## y_rep[273]  160.28    0.14 5.58  153.23  160.12  167.56  1645 1.00
## y_rep[274]  160.55    0.13 5.43  153.53  160.59  167.46  1861 1.00
## y_rep[275]  149.30    0.13 5.60  142.16  149.33  156.36  1988 1.00
## y_rep[276]  160.41    0.12 5.51  153.23  160.47  167.52  2025 1.00
## y_rep[277]  149.64    0.12 5.61  142.33  149.73  156.92  2075 1.00
## y_rep[278]  149.43    0.13 5.50  142.43  149.37  156.39  1714 1.00
## y_rep[279]  160.43    0.13 5.65  153.24  160.54  167.59  1944 1.00
## y_rep[280]  149.60    0.14 5.61  142.49  149.64  156.72  1673 1.00
## y_rep[281]  149.72    0.13 5.61  142.53  149.64  156.74  1921 1.00
## y_rep[282]  160.06    0.12 5.59  152.93  160.15  167.36  2134 1.00
## y_rep[283]  149.42    0.13 5.62  142.20  149.47  156.68  1990 1.00
## y_rep[284]  160.18    0.13 5.61  153.15  160.09  167.31  1994 1.00
## y_rep[285]  149.73    0.12 5.56  142.68  149.59  157.12  2012 1.00
## y_rep[286]  160.28    0.13 5.69  153.19  160.14  167.65  1983 1.00
## y_rep[287]  160.22    0.12 5.42  153.53  160.06  167.18  1952 1.00
## y_rep[288]  160.31    0.13 5.57  153.18  160.32  167.50  1774 1.00
## y_rep[289]  149.63    0.13 5.59  142.49  149.60  156.93  1876 1.00
## y_rep[290]  160.39    0.13 5.52  153.19  160.38  167.27  1749 1.00
## y_rep[291]  160.13    0.12 5.53  153.06  160.02  167.23  2082 1.00
## y_rep[292]  160.28    0.13 5.50  153.32  160.48  167.14  1932 1.00
## y_rep[293]  160.27    0.14 5.65  152.86  160.55  167.25  1529 1.00
## y_rep[294]  149.43    0.13 5.56  142.08  149.40  156.36  1959 1.00
## y_rep[295]  160.33    0.13 5.61  153.15  160.51  167.68  2006 1.00
## y_rep[296]  149.55    0.13 5.65  142.05  149.59  156.71  1958 1.00
## y_rep[297]  149.77    0.13 5.47  142.64  149.88  156.54  1659 1.00
## y_rep[298]  149.48    0.13 5.40  142.63  149.46  156.17  1844 1.00
## y_rep[299]  160.36    0.13 5.74  153.23  160.38  167.89  2009 1.00
## y_rep[300]  160.42    0.13 5.50  153.31  160.53  167.40  1786 1.00
## y_rep[301]  149.54    0.13 5.59  142.63  149.52  156.66  1889 1.00
## y_rep[302]  160.12    0.12 5.65  153.11  160.00  167.38  2093 1.00
## y_rep[303]  149.53    0.13 5.51  142.62  149.45  156.43  1887 1.00
## y_rep[304]  160.32    0.13 5.70  153.00  160.36  167.71  1790 1.00
## y_rep[305]  160.52    0.12 5.44  153.36  160.69  167.62  1935 1.00
## y_rep[306]  160.29    0.13 5.77  152.70  160.32  168.06  1996 1.00
## y_rep[307]  160.51    0.13 5.61  153.32  160.52  167.51  1899 1.00
## y_rep[308]  149.87    0.12 5.55  142.89  149.79  157.07  2048 1.00
## y_rep[309]  149.69    0.14 5.74  142.32  149.65  157.07  1712 1.00
## y_rep[310]  149.64    0.13 5.54  142.38  149.62  156.99  1843 1.00
## y_rep[311]  149.49    0.12 5.60  142.42  149.41  156.65  2115 1.00
## y_rep[312]  149.49    0.13 5.61  142.45  149.50  156.72  1823 1.00
## y_rep[313]  160.33    0.12 5.45  153.44  160.22  167.49  1983 1.00
## y_rep[314]  149.20    0.13 5.62  142.10  148.91  156.38  1977 1.00
## y_rep[315]  160.51    0.13 5.62  153.28  160.53  167.81  1848 1.00
## y_rep[316]  149.55    0.14 5.64  142.29  149.44  156.87  1591 1.00
## y_rep[317]  160.15    0.12 5.70  152.74  160.34  167.23  2116 1.00
## y_rep[318]  149.50    0.13 5.62  141.98  149.44  156.54  1835 1.00
## y_rep[319]  160.22    0.13 5.46  152.96  160.29  167.18  1797 1.00
## y_rep[320]  149.58    0.13 5.66  142.42  149.47  157.05  1986 1.00
## y_rep[321]  160.46    0.13 5.70  153.24  160.58  167.83  1884 1.00
## y_rep[322]  149.60    0.12 5.58  142.52  149.69  156.70  2077 1.00
## y_rep[323]  160.38    0.12 5.59  153.16  160.34  167.58  1999 1.00
## y_rep[324]  160.20    0.12 5.56  153.19  160.14  167.30  2045 1.00
## y_rep[325]  149.70    0.12 5.48  142.61  149.67  156.82  1928 1.00
## y_rep[326]  149.49    0.12 5.67  142.30  149.48  156.72  2093 1.00
## y_rep[327]  149.57    0.12 5.52  142.38  149.66  156.84  2062 1.00
## y_rep[328]  160.43    0.14 5.52  153.50  160.40  167.54  1547 1.00
## y_rep[329]  160.31    0.12 5.42  153.27  160.32  167.17  2082 1.00
## y_rep[330]  159.97    0.13 5.58  153.02  159.92  167.11  1819 1.00
## y_rep[331]  160.30    0.13 5.59  153.04  160.47  167.41  1934 1.00
## y_rep[332]  160.19    0.13 5.65  152.71  160.24  167.22  1979 1.00
## y_rep[333]  149.50    0.13 5.56  142.24  149.57  156.55  1954 1.00
## y_rep[334]  160.26    0.12 5.53  153.04  160.31  167.24  2136 1.00
## y_rep[335]  149.60    0.13 5.64  142.49  149.75  156.75  2001 1.00
## y_rep[336]  160.17    0.13 5.58  153.11  160.17  167.55  1831 1.00
## y_rep[337]  149.55    0.14 5.60  142.43  149.68  156.70  1607 1.00
## y_rep[338]  149.51    0.13 5.62  142.08  149.59  156.70  1970 1.00
## y_rep[339]  149.62    0.12 5.47  142.78  149.62  156.67  2217 1.00
## y_rep[340]  149.52    0.13 5.56  142.78  149.41  156.87  1947 1.00
## y_rep[341]  149.40    0.12 5.49  142.41  149.31  156.47  2024 1.00
## y_rep[342]  160.21    0.12 5.60  153.18  160.15  167.43  2083 1.00
## y_rep[343]  149.59    0.13 5.74  142.35  149.62  156.70  2076 1.00
## y_rep[344]  160.42    0.12 5.50  153.53  160.25  167.42  2095 1.00
## y_rep[345]  149.39    0.13 5.63  142.10  149.46  156.51  1931 1.00
## y_rep[346]  160.21    0.12 5.55  153.01  160.32  167.34  2044 1.00
## y_rep[347]  149.53    0.13 5.71  142.15  149.48  156.91  1867 1.00
## y_rep[348]  149.82    0.13 5.71  142.54  149.87  157.28  1963 1.00
## y_rep[349]  149.40    0.13 5.62  142.24  149.45  156.67  1982 1.00
## y_rep[350]  160.23    0.13 5.65  152.82  160.29  167.52  1857 1.00
## y_rep[351]  149.41    0.12 5.58  142.42  149.37  156.56  2084 1.00
## y_rep[352]  160.32    0.12 5.58  153.37  160.07  167.40  2011 1.00
## lp__       -782.25    0.06 1.24 -783.95 -781.93 -781.05   393 1.01
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 15:25:48 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

5 Polynomial regression

5.1 Howell 2: subset with only individuals over 18 years old

Standardizing the explanatory variable:

howell2 <- howell2 %>% mutate(weight.z = (weight - mean(weight)) / sd(weight))

Data in a list:

m6.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.z)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}

parameters {                                               // unobserved variables
  real beta[3];                                            // parameters: intercept and slope
  real sigma;
}

model {
  height ~ normal(beta[1] + beta[2]*weight + beta[3]*(weight .* weight),sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ normal(0, 100);                                  // prior of the slope
  beta[3] ~ normal(0, 100);
  sigma ~ uniform(0,50);
}

generated quantities {
  vector[N] y_rep;

  for(i in 1:N)
  y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i] + beta[3]*(weight[i] .* weight[i]), sigma);
}
fit.m6 <- sampling(m6, data = m6.data, iter = 1000, chains = 4, cores = 4)
print(fit.m6, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: c7fbeba14f23c52987bbbb1ee94a3a29.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##               mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]     154.60    0.01 0.36  154.16  154.60  155.06  1418 1.00
## beta[2]       5.86    0.01 0.27    5.51    5.86    6.20   469 1.01
## beta[3]      -0.01    0.01 0.22   -0.30   -0.01    0.28  1421 1.00
## sigma         5.12    0.00 0.19    4.88    5.11    5.37  1951 1.00
## y_rep[1]    156.96    0.11 5.20  150.21  156.93  163.47  2056 1.00
## y_rep[2]    147.11    0.12 5.06  140.63  146.95  153.80  1768 1.00
## y_rep[3]    142.68    0.12 5.09  136.05  142.78  149.08  1912 1.00
## y_rep[4]    161.82    0.12 5.12  155.35  161.80  168.23  1837 1.00
## y_rep[5]    150.86    0.12 5.14  144.33  150.86  157.45  1960 1.00
## y_rep[6]    170.89    0.12 5.40  164.17  170.89  177.88  2034 1.00
## y_rep[7]    148.39    0.12 5.01  142.14  148.27  154.98  1658 1.00
## y_rep[8]    163.92    0.11 5.15  157.14  164.08  170.35  2110 1.00
## y_rep[9]    145.49    0.11 5.00  139.05  145.48  152.13  1918 1.00
## y_rep[10]   163.15    0.11 5.18  156.73  163.13  169.61  2070 1.00
## y_rep[11]   158.92    0.12 5.19  152.48  158.89  165.58  1990 1.00
## y_rep[12]   150.99    0.11 5.12  144.64  150.84  157.53  2035 1.00
## y_rep[13]   146.40    0.11 4.99  140.12  146.31  152.88  1978 1.00
## y_rep[14]   156.98    0.12 5.13  150.59  156.91  163.32  1945 1.00
## y_rep[15]   144.44    0.12 5.25  137.79  144.39  151.24  1907 1.00
## y_rep[16]   157.84    0.12 5.26  151.34  157.68  164.85  1873 1.00
## y_rep[17]   152.19    0.11 5.14  145.74  152.13  158.73  2086 1.00
## y_rep[18]   148.45    0.12 5.31  141.64  148.36  155.35  2009 1.00
## y_rep[19]   158.21    0.11 5.07  151.85  158.30  164.68  2052 1.00
## y_rep[20]   152.70    0.12 5.10  146.16  152.72  159.11  1901 1.00
## y_rep[21]   145.96    0.12 5.17  139.18  146.05  152.52  1737 1.00
## y_rep[22]   148.11    0.11 5.23  141.44  148.18  154.60  2070 1.00
## y_rep[23]   145.87    0.12 5.07  139.27  145.81  152.32  1839 1.00
## y_rep[24]   150.47    0.11 5.18  143.79  150.53  157.11  2039 1.00
## y_rep[25]   163.75    0.11 5.02  157.31  163.77  170.14  2065 1.00
## y_rep[26]   148.04    0.12 5.20  141.46  148.07  154.90  2001 1.00
## y_rep[27]   148.57    0.11 5.06  141.96  148.51  155.05  1998 1.00
## y_rep[28]   155.99    0.11 5.19  149.44  155.92  162.81  2201 1.00
## y_rep[29]   148.92    0.12 5.06  142.40  148.95  155.35  1889 1.00
## y_rep[30]   146.05    0.11 5.09  139.68  145.96  152.63  2089 1.00
## y_rep[31]   156.79    0.11 5.11  150.45  156.68  163.49  2039 1.00
## y_rep[32]   157.19    0.12 5.15  150.49  157.24  163.64  1927 1.00
## y_rep[33]   158.62    0.11 5.00  152.17  158.63  164.98  2029 1.00
## y_rep[34]   158.63    0.11 5.04  152.09  158.65  165.11  2043 1.00
## y_rep[35]   165.21    0.12 5.21  158.61  165.26  172.11  1976 1.00
## y_rep[36]   149.40    0.12 5.01  142.77  149.37  155.93  1832 1.00
## y_rep[37]   159.05    0.11 5.16  152.34  159.06  165.79  2099 1.00
## y_rep[38]   151.07    0.11 5.19  144.59  150.92  157.83  2169 1.00
## y_rep[39]   148.84    0.12 5.07  142.29  148.78  155.47  1915 1.00
## y_rep[40]   152.16    0.11 5.02  145.87  152.03  158.70  2008 1.00
## y_rep[41]   154.28    0.12 5.25  147.52  154.28  160.79  2079 1.00
## y_rep[42]   166.83    0.12 5.32  159.95  166.90  173.46  1930 1.00
## y_rep[43]   156.14    0.11 5.19  149.57  156.08  162.90  2045 1.00
## y_rep[44]   153.85    0.11 5.14  147.45  153.90  160.43  2126 1.00
## y_rep[45]   160.00    0.12 5.15  153.31  160.16  166.46  1905 1.00
## y_rep[46]   163.00    0.10 4.91  156.62  162.96  169.28  2227 1.00
## y_rep[47]   155.36    0.12 5.11  148.55  155.34  161.75  1789 1.00
## y_rep[48]   157.51    0.11 5.05  151.27  157.36  163.92  2036 1.00
## y_rep[49]   161.09    0.12 4.98  154.87  161.09  167.53  1805 1.00
## y_rep[50]   156.89    0.11 5.17  150.22  156.76  163.59  2086 1.00
## y_rep[51]   155.18    0.12 5.10  148.78  155.07  161.78  1894 1.00
## y_rep[52]   152.13    0.12 5.19  145.59  152.17  158.76  1896 1.00
## y_rep[53]   146.81    0.12 5.12  140.24  146.84  153.34  1970 1.00
## y_rep[54]   164.54    0.11 5.07  158.08  164.45  171.03  2024 1.00
## y_rep[55]   147.55    0.11 4.99  141.23  147.50  153.99  2034 1.00
## y_rep[56]   157.54    0.11 5.03  150.96  157.53  163.99  2123 1.00
## y_rep[57]   147.72    0.12 5.02  141.31  147.68  154.08  1803 1.00
## y_rep[58]   156.62    0.11 5.03  150.20  156.69  163.05  2055 1.00
## y_rep[59]   156.65    0.11 5.05  150.33  156.52  163.09  2087 1.00
## y_rep[60]   165.93    0.11 5.28  159.32  165.83  172.71  2108 1.00
## y_rep[61]   148.15    0.12 5.26  141.61  148.11  154.83  2088 1.00
## y_rep[62]   156.57    0.12 5.18  150.13  156.53  163.20  1918 1.00
## y_rep[63]   152.80    0.12 5.04  146.28  152.92  159.17  1904 1.00
## y_rep[64]   159.33    0.12 5.15  152.68  159.30  165.78  1885 1.00
## y_rep[65]   149.37    0.11 5.07  142.87  149.34  155.75  2005 1.00
## y_rep[66]   159.60    0.11 5.17  153.00  159.61  166.33  2035 1.00
## y_rep[67]   149.78    0.12 5.34  142.96  149.92  156.46  2079 1.00
## y_rep[68]   146.69    0.12 5.02  140.22  146.64  153.13  1679 1.00
## y_rep[69]   155.69    0.11 5.08  148.98  155.71  162.27  2049 1.00
## y_rep[70]   152.34    0.12 5.26  145.56  152.32  159.02  1853 1.00
## y_rep[71]   157.21    0.11 4.99  151.15  157.06  163.66  2007 1.00
## y_rep[72]   164.37    0.12 5.15  157.83  164.26  171.10  1927 1.00
## y_rep[73]   157.91    0.12 5.15  151.36  157.81  164.42  1822 1.00
## y_rep[74]   152.64    0.11 5.09  146.18  152.59  159.16  2126 1.00
## y_rep[75]   145.58    0.11 5.11  139.16  145.50  152.21  1993 1.00
## y_rep[76]   165.24    0.12 5.23  158.52  165.32  172.02  2007 1.00
## y_rep[77]   160.08    0.11 5.12  153.48  159.94  166.74  2010 1.00
## y_rep[78]   156.67    0.12 5.17  150.15  156.66  163.15  1836 1.00
## y_rep[79]   151.12    0.11 5.08  144.74  151.06  157.61  2004 1.00
## y_rep[80]   143.58    0.12 5.09  137.16  143.50  150.13  1918 1.00
## y_rep[81]   165.34    0.13 5.29  158.53  165.32  171.96  1732 1.00
## y_rep[82]   152.78    0.12 5.11  146.60  152.62  159.14  1950 1.00
## y_rep[83]   150.18    0.12 5.14  143.60  150.04  157.04  1841 1.00
## y_rep[84]   155.69    0.11 5.08  149.38  155.69  162.26  2017 1.00
## y_rep[85]   151.09    0.12 5.07  144.59  151.02  157.48  1855 1.00
## y_rep[86]   157.00    0.12 5.19  150.33  156.96  163.71  1868 1.00
## y_rep[87]   147.79    0.12 5.23  141.19  147.88  154.46  1980 1.00
## y_rep[88]   154.55    0.12 5.25  147.95  154.51  161.07  1994 1.00
## y_rep[89]   151.91    0.12 5.15  145.45  151.89  158.69  1980 1.00
## y_rep[90]   163.55    0.11 5.13  156.95  163.59  170.06  2126 1.00
## y_rep[91]   162.47    0.11 5.12  156.13  162.33  169.18  2134 1.00
## y_rep[92]   161.14    0.12 5.10  154.58  161.04  167.99  1927 1.00
## y_rep[93]   152.15    0.12 5.18  145.50  152.00  158.90  1925 1.00
## y_rep[94]   158.69    0.11 5.07  151.95  158.74  165.03  2012 1.00
## y_rep[95]   151.62    0.12 5.08  145.09  151.75  158.13  1944 1.00
## y_rep[96]   164.62    0.12 5.24  157.82  164.61  171.40  2009 1.00
## y_rep[97]   152.20    0.12 5.24  145.43  152.13  158.88  1860 1.00
## y_rep[98]   162.04    0.12 5.17  155.39  162.04  168.46  1879 1.00
## y_rep[99]   159.37    0.12 5.25  152.39  159.40  165.83  1980 1.00
## y_rep[100]  152.33    0.11 5.08  145.82  152.35  158.86  1953 1.00
## y_rep[101]  158.31    0.11 5.07  152.00  158.19  164.93  1992 1.00
## y_rep[102]  148.71    0.11 5.08  142.06  148.77  155.20  2134 1.00
## y_rep[103]  158.91    0.11 5.06  152.45  158.85  165.69  2006 1.00
## y_rep[104]  167.83    0.13 5.24  161.30  167.85  174.52  1728 1.00
## y_rep[105]  157.08    0.12 5.06  150.62  157.12  163.56  1932 1.00
## y_rep[106]  149.22    0.11 5.06  142.61  149.34  155.58  2095 1.00
## y_rep[107]  147.09    0.11 5.10  140.47  147.08  153.56  2177 1.00
## y_rep[108]  151.98    0.12 5.11  145.60  151.98  158.57  1957 1.00
## y_rep[109]  154.16    0.11 5.21  147.70  154.23  160.57  2167 1.00
## y_rep[110]  157.26    0.11 5.08  150.65  157.35  163.63  2012 1.00
## y_rep[111]  163.87    0.11 5.12  157.32  163.96  170.45  2074 1.00
## y_rep[112]  143.87    0.12 5.21  137.06  143.87  150.55  2018 1.00
## y_rep[113]  150.89    0.12 5.24  144.30  150.94  157.65  1992 1.00
## y_rep[114]  160.28    0.12 5.14  153.85  160.30  166.70  1964 1.00
## y_rep[115]  158.13    0.12 5.12  151.70  158.07  164.58  1874 1.00
## y_rep[116]  162.20    0.12 5.12  155.57  162.31  168.60  1746 1.00
## y_rep[117]  162.79    0.13 5.25  155.94  162.98  169.45  1752 1.00
## y_rep[118]  163.97    0.12 5.17  157.31  164.00  170.60  1868 1.00
## y_rep[119]  152.08    0.12 5.16  145.45  151.95  158.55  2011 1.00
## y_rep[120]  150.96    0.12 5.12  144.27  150.98  157.52  1970 1.00
## y_rep[121]  158.61    0.12 5.15  151.82  158.59  165.30  1892 1.00
## y_rep[122]  154.12    0.11 5.07  147.48  154.18  160.72  1983 1.00
## y_rep[123]  155.46    0.12 5.16  148.64  155.50  162.10  1762 1.00
## y_rep[124]  151.67    0.13 5.22  145.00  151.55  158.36  1662 1.00
## y_rep[125]  160.49    0.11 5.01  154.15  160.45  167.07  1962 1.00
## y_rep[126]  153.94    0.12 5.23  147.45  154.02  160.54  2014 1.00
## y_rep[127]  142.89    0.12 5.21  136.26  142.94  149.50  1949 1.00
## y_rep[128]  165.04    0.12 5.17  158.45  165.09  171.34  1928 1.00
## y_rep[129]  161.60    0.12 5.21  154.86  161.56  168.20  1922 1.00
## y_rep[130]  146.82    0.12 5.25  140.05  147.02  153.45  1907 1.00
## y_rep[131]  158.14    0.13 5.20  151.34  158.05  164.74  1686 1.00
## y_rep[132]  165.29    0.12 5.15  158.68  165.32  171.99  1823 1.00
## y_rep[133]  152.17    0.11 5.20  145.51  152.15  158.71  2147 1.00
## y_rep[134]  159.36    0.12 5.19  152.61  159.30  166.09  1740 1.00
## y_rep[135]  156.23    0.11 5.08  149.91  156.13  162.56  2011 1.00
## y_rep[136]  169.63    0.12 5.30  162.69  169.70  176.25  1888 1.00
## y_rep[137]  158.52    0.11 5.03  151.91  158.60  164.99  2037 1.00
## y_rep[138]  142.39    0.11 5.14  135.71  142.38  149.20  2029 1.00
## y_rep[139]  145.64    0.11 5.17  139.19  145.57  152.22  2101 1.00
## y_rep[140]  155.09    0.12 4.98  148.81  155.05  161.49  1789 1.00
## y_rep[141]  153.84    0.13 5.20  147.21  153.73  160.67  1708 1.00
## y_rep[142]  155.10    0.12 5.18  148.28  155.32  161.87  1904 1.00
## y_rep[143]  149.30    0.15 5.13  142.91  149.32  155.83  1206 1.00
## y_rep[144]  150.94    0.11 5.00  144.52  150.87  157.33  1901 1.00
## y_rep[145]  154.86    0.11 5.11  148.34  154.84  161.33  2107 1.00
## y_rep[146]  162.61    0.12 5.27  155.89  162.61  169.35  1913 1.00
## y_rep[147]  161.24    0.12 5.15  154.51  161.16  167.91  1866 1.00
## y_rep[148]  149.25    0.11 5.15  142.54  149.26  155.82  2054 1.00
## y_rep[149]  155.13    0.11 5.15  148.60  155.23  161.80  2042 1.00
## y_rep[150]  150.41    0.11 5.04  144.14  150.34  156.89  1977 1.00
## y_rep[151]  160.60    0.12 5.02  153.95  160.65  167.13  1904 1.00
## y_rep[152]  148.95    0.12 5.19  142.40  148.94  155.38  2025 1.00
## y_rep[153]  149.40    0.11 5.14  142.87  149.50  155.99  1999 1.00
## y_rep[154]  153.86    0.12 5.17  147.19  153.85  160.59  1880 1.00
## y_rep[155]  149.86    0.13 5.25  143.22  149.87  156.64  1620 1.00
## y_rep[156]  141.82    0.12 5.17  135.27  141.88  148.46  1954 1.00
## y_rep[157]  156.20    0.12 5.09  149.56  156.15  162.79  1927 1.00
## y_rep[158]  150.59    0.11 4.96  144.03  150.52  156.86  1989 1.00
## y_rep[159]  159.29    0.11 5.02  152.85  159.26  165.78  2038 1.00
## y_rep[160]  151.13    0.12 5.12  144.58  151.02  157.80  1724 1.00
## y_rep[161]  163.22    0.12 5.14  156.58  163.40  169.77  1788 1.00
## y_rep[162]  154.74    0.12 5.08  147.99  154.80  161.23  1783 1.00
## y_rep[163]  146.22    0.13 5.20  139.38  146.29  152.75  1661 1.00
## y_rep[164]  154.84    0.12 5.20  148.11  154.74  161.47  1976 1.00
## y_rep[165]  157.37    0.11 5.09  150.97  157.39  163.75  2089 1.00
## y_rep[166]  155.13    0.12 5.17  148.27  155.18  161.84  1913 1.00
## y_rep[167]  157.50    0.12 5.16  150.91  157.58  164.09  1880 1.00
## y_rep[168]  151.07    0.12 5.24  144.04  151.08  157.66  2000 1.00
## y_rep[169]  148.75    0.11 4.99  142.25  148.79  155.01  2039 1.00
## y_rep[170]  152.06    0.12 5.11  145.65  151.83  158.58  1912 1.00
## y_rep[171]  148.72    0.12 5.22  142.20  148.80  155.38  1995 1.00
## y_rep[172]  157.52    0.11 5.11  151.18  157.49  164.12  2007 1.00
## y_rep[173]  149.02    0.12 5.16  142.51  148.92  155.66  2001 1.00
## y_rep[174]  150.09    0.12 5.04  143.69  149.96  156.73  1700 1.00
## y_rep[175]  159.24    0.11 4.92  152.71  159.33  165.57  2008 1.00
## y_rep[176]  162.93    0.12 5.05  156.55  163.14  169.21  1918 1.00
## y_rep[177]  154.97    0.11 5.28  148.24  155.03  161.58  2144 1.00
## y_rep[178]  149.68    0.12 5.25  142.92  149.62  156.29  1791 1.00
## y_rep[179]  152.75    0.12 5.28  146.07  152.68  159.51  1995 1.00
## y_rep[180]  142.56    0.12 5.33  135.57  142.57  149.48  1821 1.00
## y_rep[181]  155.13    0.11 5.09  148.65  155.03  161.68  1997 1.00
## y_rep[182]  148.35    0.12 5.20  141.58  148.45  154.85  2021 1.00
## y_rep[183]  146.44    0.12 5.14  139.92  146.42  152.87  1859 1.00
## y_rep[184]  151.02    0.12 5.07  144.49  151.06  157.54  1917 1.00
## y_rep[185]  157.17    0.12 5.12  150.52  157.26  163.63  1892 1.00
## y_rep[186]  157.01    0.12 5.14  150.64  157.11  163.34  1811 1.00
## y_rep[187]  155.91    0.12 5.15  149.14  156.00  162.46  1838 1.00
## y_rep[188]  151.15    0.11 5.11  144.60  151.25  157.67  2153 1.00
## y_rep[189]  146.98    0.12 5.06  140.12  147.03  153.16  1911 1.00
## y_rep[190]  157.29    0.13 5.10  150.55  157.39  163.70  1520 1.00
## y_rep[191]  147.07    0.12 5.13  140.61  147.05  153.64  1978 1.00
## y_rep[192]  164.78    0.12 5.25  158.04  164.87  171.53  2041 1.00
## y_rep[193]  149.01    0.12 5.13  142.51  148.97  155.65  1925 1.00
## y_rep[194]  156.86    0.12 5.27  149.95  156.96  163.59  1927 1.00
## y_rep[195]  146.75    0.13 5.20  140.15  146.68  153.32  1715 1.00
## y_rep[196]  151.38    0.12 5.13  144.66  151.47  157.95  1977 1.00
## y_rep[197]  154.55    0.12 5.17  147.91  154.60  161.12  1985 1.00
## y_rep[198]  159.47    0.12 5.19  152.55  159.62  166.02  1820 1.00
## y_rep[199]  164.04    0.11 5.14  157.85  163.93  170.47  2042 1.00
## y_rep[200]  148.16    0.11 5.05  141.63  148.30  154.63  1931 1.00
## y_rep[201]  166.64    0.12 5.24  159.98  166.68  173.24  1966 1.00
## y_rep[202]  154.26    0.11 5.06  147.89  154.26  160.68  1940 1.00
## y_rep[203]  153.97    0.12 5.23  147.28  154.02  160.68  1934 1.00
## y_rep[204]  154.09    0.13 5.10  147.38  154.15  160.31  1595 1.00
## y_rep[205]  154.10    0.12 5.17  147.37  154.14  160.74  1992 1.00
## y_rep[206]  150.49    0.11 5.09  144.16  150.44  157.25  2074 1.00
## y_rep[207]  153.74    0.11 5.06  147.23  153.78  160.42  1976 1.00
## y_rep[208]  144.94    0.13 5.23  138.48  144.76  151.72  1593 1.00
## y_rep[209]  143.48    0.13 5.19  136.98  143.34  150.15  1654 1.00
## y_rep[210]  146.36    0.12 5.18  139.77  146.41  152.82  1997 1.00
## y_rep[211]  158.77    0.12 5.18  152.23  158.91  165.43  1847 1.00
## y_rep[212]  161.92    0.12 5.05  155.34  161.97  168.42  1823 1.00
## y_rep[213]  147.38    0.11 5.23  140.48  147.51  154.06  2113 1.00
## y_rep[214]  150.57    0.11 5.13  144.04  150.57  157.19  2004 1.00
## y_rep[215]  159.73    0.11 4.93  153.15  159.83  166.01  1921 1.00
## y_rep[216]  153.68    0.12 5.18  147.07  153.73  160.41  1861 1.00
## y_rep[217]  158.96    0.12 5.15  152.46  158.93  165.69  1752 1.00
## y_rep[218]  144.38    0.11 5.22  137.80  144.50  151.08  2121 1.00
## y_rep[219]  153.54    0.12 5.13  146.80  153.65  160.03  1936 1.00
## y_rep[220]  157.49    0.12 5.19  150.82  157.67  164.12  1996 1.00
## y_rep[221]  152.78    0.12 5.12  146.35  152.70  159.57  1933 1.00
## y_rep[222]  144.66    0.12 5.20  138.22  144.62  151.36  1979 1.00
## y_rep[223]  149.83    0.12 5.18  143.03  150.07  156.56  1823 1.00
## y_rep[224]  146.20    0.11 5.07  139.64  146.25  152.62  2118 1.00
## y_rep[225]  160.02    0.12 5.09  153.43  159.97  166.44  1809 1.00
## y_rep[226]  155.23    0.11 5.15  148.59  155.22  161.84  2032 1.00
## y_rep[227]  158.21    0.11 5.22  151.31  158.29  164.92  2080 1.00
## y_rep[228]  167.11    0.12 5.19  160.68  166.90  173.85  1763 1.00
## y_rep[229]  153.46    0.12 5.17  146.98  153.42  160.04  1772 1.00
## y_rep[230]  161.04    0.11 5.21  154.30  160.98  167.69  2188 1.00
## y_rep[231]  159.84    0.12 5.04  153.16  160.07  166.23  1882 1.00
## y_rep[232]  144.86    0.11 5.08  138.29  144.80  151.38  1972 1.00
## y_rep[233]  149.45    0.11 5.09  143.03  149.44  155.82  1997 1.00
## y_rep[234]  167.68    0.12 5.36  160.84  167.60  174.66  2134 1.00
## y_rep[235]  161.02    0.12 5.25  154.41  161.05  167.74  1937 1.00
## y_rep[236]  150.03    0.12 5.12  143.58  149.94  156.66  1971 1.00
## y_rep[237]  153.71    0.11 5.10  147.40  153.61  160.28  1997 1.00
## y_rep[238]  163.29    0.11 5.13  156.58  163.43  169.89  2030 1.00
## y_rep[239]  154.74    0.12 5.20  148.10  154.61  161.56  1947 1.00
## y_rep[240]  151.20    0.12 5.18  144.64  151.22  157.75  1925 1.00
## y_rep[241]  151.54    0.12 5.17  144.98  151.43  158.21  1943 1.00
## y_rep[242]  152.41    0.11 5.04  146.04  152.29  158.93  2087 1.00
## y_rep[243]  152.94    0.11 5.01  146.78  152.85  159.29  2023 1.00
## y_rep[244]  150.85    0.11 5.09  144.28  150.99  157.50  2100 1.00
## y_rep[245]  152.94    0.12 5.26  146.16  152.85  159.76  1983 1.00
## y_rep[246]  155.72    0.12 5.28  148.91  155.78  162.31  1979 1.00
## y_rep[247]  157.09    0.11 5.06  150.56  157.12  163.49  2042 1.00
## y_rep[248]  152.35    0.12 5.21  145.83  152.38  158.97  1890 1.00
## y_rep[249]  157.76    0.12 5.18  151.11  157.69  164.52  1916 1.00
## y_rep[250]  155.46    0.12 5.08  149.06  155.53  161.70  1830 1.00
## y_rep[251]  161.73    0.12 5.20  154.96  161.80  168.36  1951 1.00
## y_rep[252]  156.93    0.12 5.11  150.42  156.89  163.45  1784 1.00
## y_rep[253]  153.44    0.11 5.01  146.91  153.40  160.05  2083 1.00
## y_rep[254]  153.16    0.11 5.07  146.57  153.25  159.37  1948 1.00
## y_rep[255]  161.84    0.11 5.13  155.21  161.73  168.39  2020 1.00
## y_rep[256]  154.57    0.12 5.29  147.88  154.50  161.46  1835 1.00
## y_rep[257]  156.59    0.12 5.13  150.07  156.75  163.03  1953 1.00
## y_rep[258]  142.51    0.13 5.32  135.75  142.49  149.22  1667 1.00
## y_rep[259]  164.74    0.12 5.19  157.94  164.86  171.16  1894 1.00
## y_rep[260]  164.47    0.11 5.08  158.03  164.47  170.87  1980 1.00
## y_rep[261]  157.17    0.12 5.17  150.79  157.09  163.75  1935 1.00
## y_rep[262]  154.37    0.11 5.06  147.73  154.39  160.93  2076 1.00
## y_rep[263]  155.00    0.11 5.12  148.62  155.00  161.58  1987 1.00
## y_rep[264]  158.11    0.12 5.11  151.60  158.11  164.64  1791 1.00
## y_rep[265]  156.03    0.11 5.09  149.44  155.96  162.31  1961 1.00
## y_rep[266]  153.42    0.12 5.18  146.63  153.52  160.00  1820 1.00
## y_rep[267]  164.64    0.12 5.20  157.78  164.67  171.35  2038 1.00
## y_rep[268]  150.26    0.12 5.12  143.67  150.20  156.91  1933 1.00
## y_rep[269]  160.13    0.11 5.01  153.65  160.11  166.37  1962 1.00
## y_rep[270]  155.73    0.12 5.09  149.21  155.84  162.11  1958 1.00
## y_rep[271]  158.33    0.13 5.13  151.85  158.37  164.99  1684 1.00
## y_rep[272]  153.91    0.12 5.06  147.45  153.84  160.60  1718 1.00
## y_rep[273]  160.20    0.12 5.10  153.66  160.19  166.77  1931 1.00
## y_rep[274]  150.66    0.12 5.23  144.02  150.59  157.36  2064 1.00
## y_rep[275]  147.61    0.11 5.13  140.98  147.71  154.22  2031 1.00
## y_rep[276]  160.59    0.12 5.10  154.02  160.60  167.07  1791 1.00
## y_rep[277]  154.34    0.12 5.14  147.87  154.37  160.79  1870 1.00
## y_rep[278]  153.17    0.12 5.19  146.58  153.14  159.78  1937 1.00
## y_rep[279]  155.99    0.12 5.18  149.41  156.01  162.86  1828 1.00
## y_rep[280]  149.67    0.11 4.99  143.32  149.73  155.98  1935 1.00
## y_rep[281]  145.42    0.12 5.11  138.90  145.44  152.00  1919 1.00
## y_rep[282]  149.49    0.11 5.14  142.76  149.41  156.24  2103 1.00
## y_rep[283]  147.80    0.14 5.12  141.18  147.74  154.48  1338 1.00
## y_rep[284]  157.74    0.13 5.33  150.97  157.87  164.48  1808 1.00
## y_rep[285]  146.25    0.12 5.04  139.67  146.35  152.59  1721 1.00
## y_rep[286]  153.48    0.12 5.12  146.83  153.45  159.91  1810 1.00
## y_rep[287]  157.52    0.11 5.24  150.84  157.42  164.11  2248 1.00
## y_rep[288]  160.10    0.11 5.01  153.83  160.13  166.35  1971 1.00
## y_rep[289]  153.83    0.11 5.10  147.46  153.73  160.38  2030 1.00
## y_rep[290]  157.45    0.11 5.06  151.02  157.40  163.97  2290 1.00
## y_rep[291]  156.19    0.12 5.20  149.57  156.21  162.92  1899 1.00
## y_rep[292]  165.03    0.12 5.12  158.55  165.08  171.55  1877 1.00
## y_rep[293]  157.84    0.11 5.11  151.39  157.62  164.46  2009 1.00
## y_rep[294]  157.69    0.11 5.03  151.25  157.80  164.08  2081 1.00
## y_rep[295]  156.93    0.11 5.01  150.51  156.86  163.24  2156 1.00
## y_rep[296]  146.00    0.12 5.23  139.56  146.02  152.80  1934 1.00
## y_rep[297]  147.02    0.11 5.00  140.61  147.02  153.34  1937 1.00
## y_rep[298]  158.23    0.11 5.03  151.61  158.25  164.74  1990 1.00
## y_rep[299]  160.19    0.12 5.03  153.86  160.13  166.75  1867 1.00
## y_rep[300]  156.12    0.12 5.28  149.35  155.94  162.87  1980 1.00
## y_rep[301]  147.35    0.12 5.09  140.91  147.28  153.80  1745 1.00
## y_rep[302]  157.90    0.12 5.17  151.32  157.86  164.67  1970 1.00
## y_rep[303]  148.32    0.11 5.10  141.94  148.33  155.12  2059 1.00
## y_rep[304]  156.74    0.11 5.15  150.02  156.77  163.58  2094 1.00
## y_rep[305]  164.10    0.12 5.25  157.22  164.10  170.65  1855 1.00
## y_rep[306]  163.00    0.11 5.11  156.58  163.00  169.65  2052 1.00
## y_rep[307]  157.85    0.11 5.06  151.22  157.93  164.37  1948 1.00
## y_rep[308]  152.83    0.12 5.10  146.18  152.75  159.32  1918 1.00
## y_rep[309]  144.68    0.12 5.23  137.79  144.74  151.16  1920 1.00
## y_rep[310]  159.14    0.12 5.20  152.48  159.14  165.96  1928 1.00
## y_rep[311]  153.84    0.12 5.24  147.02  153.92  160.35  2030 1.00
## y_rep[312]  144.27    0.12 5.21  137.92  144.31  150.94  1946 1.00
## y_rep[313]  156.65    0.12 5.12  150.26  156.64  163.06  1751 1.00
## y_rep[314]  155.51    0.13 5.25  148.81  155.57  161.91  1537 1.00
## y_rep[315]  156.68    0.12 5.16  150.20  156.70  163.10  1864 1.00
## y_rep[316]  150.78    0.11 5.26  144.09  150.68  157.39  2232 1.00
## y_rep[317]  159.47    0.12 5.15  153.03  159.48  166.01  1851 1.00
## y_rep[318]  154.77    0.11 5.20  147.97  154.82  161.31  2227 1.00
## y_rep[319]  152.02    0.11 5.11  145.46  151.99  158.52  2026 1.00
## y_rep[320]  151.67    0.11 5.13  145.06  151.69  158.23  2106 1.00
## y_rep[321]  154.57    0.11 5.14  148.08  154.50  161.21  2133 1.00
## y_rep[322]  160.15    0.11 5.14  153.60  160.05  166.75  2021 1.00
## y_rep[323]  158.42    0.11 5.09  151.92  158.47  164.89  2025 1.00
## y_rep[324]  160.25    0.12 5.25  153.31  160.32  166.86  2022 1.00
## y_rep[325]  153.62    0.12 5.15  146.99  153.59  160.12  1979 1.00
## y_rep[326]  146.11    0.12 4.99  139.81  146.12  152.34  1861 1.00
## y_rep[327]  152.53    0.12 5.06  145.97  152.51  159.03  1699 1.00
## y_rep[328]  156.49    0.12 5.21  149.71  156.43  163.16  2035 1.00
## y_rep[329]  151.66    0.12 5.07  145.26  151.64  158.19  1887 1.00
## y_rep[330]  148.41    0.11 5.04  141.88  148.47  154.86  2041 1.00
## y_rep[331]  153.13    0.11 5.25  146.36  153.18  159.83  2084 1.00
## y_rep[332]  162.26    0.12 5.05  155.56  162.38  168.67  1836 1.00
## y_rep[333]  152.71    0.11 5.19  146.05  152.73  159.43  2124 1.00
## y_rep[334]  149.88    0.11 5.04  143.40  149.90  156.23  2053 1.00
## y_rep[335]  153.14    0.12 5.13  146.70  153.16  159.64  1945 1.00
## y_rep[336]  161.66    0.11 5.05  155.10  161.81  168.03  2212 1.00
## y_rep[337]  148.89    0.12 5.10  142.28  148.90  155.46  1936 1.00
## y_rep[338]  152.65    0.12 5.02  146.29  152.75  158.96  1869 1.00
## y_rep[339]  146.29    0.12 5.06  139.85  146.20  152.87  1915 1.00
## y_rep[340]  153.75    0.12 5.11  147.21  153.76  160.27  1804 1.00
## y_rep[341]  148.39    0.11 5.10  141.82  148.42  154.75  1998 1.00
## y_rep[342]  154.01    0.11 5.24  147.44  153.88  160.72  2117 1.00
## y_rep[343]  153.83    0.12 5.24  147.17  153.81  160.46  1972 1.00
## y_rep[344]  157.35    0.12 5.09  150.88  157.26  164.11  1894 1.00
## y_rep[345]  149.50    0.11 5.17  142.71  149.62  156.02  2078 1.00
## y_rep[346]  151.02    0.11 5.11  144.54  150.94  157.75  2151 1.00
## y_rep[347]  150.94    0.12 5.23  144.33  150.89  157.65  1849 1.00
## y_rep[348]  156.21    0.12 5.09  149.57  156.42  162.70  1795 1.00
## y_rep[349]  145.00    0.11 5.19  138.38  144.94  151.72  2052 1.00
## y_rep[350]  160.97    0.12 5.25  154.15  160.99  167.66  1988 1.00
## y_rep[351]  162.93    0.11 5.09  156.41  162.97  169.60  2032 1.00
## y_rep[352]  161.19    0.12 5.14  154.59  161.20  167.91  1942 1.00
## lp__       -749.55    0.05 1.39 -751.40 -749.20 -748.11   818 1.00
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 15:25:51 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
summary(lm(height~weight.z+I(weight.z^2),data=howell2))
## 
## Call:
## lm(formula = height ~ weight.z + I(weight.z^2), data = howell2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.7517  -2.8933   0.0313   3.1419  14.7936 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   154.60953    0.35209 439.120   <2e-16 ***
## weight.z        5.84515    0.27348  21.373   <2e-16 ***
## I(weight.z^2)  -0.01248    0.22483  -0.055    0.956    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.094 on 349 degrees of freedom
## Multiple R-squared:  0.5696, Adjusted R-squared:  0.5672 
## F-statistic:   231 on 2 and 349 DF,  p-value: < 2.2e-16

See Figure 4.9 of Statistical rethinking

https://github.com/ssp3nc3r/rethinking/blob/master/chapter04.Rmd

Here, all the variance is represented (variance of actual heights).

5.2 Howell 1: data set with all individuals

Standardizing the explanatory variable:

howell1 <- howell1 %>% mutate(weight.z = (weight - mean(weight)) / sd(weight))

Data in a list:

m7.data <- list(N = nrow(howell1), height = howell1$height, weight = howell1$weight.z)
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
}

parameters {                                               // unobserved variables
  real beta[3];                                            // parameters: intercept and slope
  real sigma;
}

model {
  height ~ normal(beta[1] + beta[2]*weight + beta[3]*(weight .* weight),sigma);        // likelihood
  beta[1] ~ normal(178, 100);                              // prior of the intercept
  beta[2] ~ normal(0, 100);                                  // prior of the slope
  beta[3] ~ normal(0, 100);
  sigma ~ uniform(0,50);
}

generated quantities {
  vector[N] y_rep;

  for(i in 1:N)
  y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i] + beta[3]*(weight[i] .* weight[i]), sigma);
}
fit.m7 <- sampling(m7, data = m7.data, iter = 1000, chains = 4, cores = 4)
print(fit.m7, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: c7fbeba14f23c52987bbbb1ee94a3a29.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##                mean se_mean   sd      10%      50%      90% n_eff Rhat
## beta[1]      146.65    0.01 0.37   146.15   146.65   147.10   833    1
## beta[2]       21.44    0.01 0.29    21.07    21.44    21.80   870    1
## beta[3]       -8.40    0.01 0.28    -8.77    -8.40    -8.05   777    1
## sigma          5.78    0.00 0.17     5.56     5.77     6.00  2177    1
## y_rep[1]     158.64    0.13 5.86   151.00   158.69   166.23  1904    1
## y_rep[2]     148.05    0.13 5.81   140.84   148.13   155.35  1983    1
## y_rep[3]     140.88    0.14 5.79   133.32   141.07   148.17  1797    1
## y_rep[4]     160.30    0.13 5.80   152.89   160.17   167.85  2078    1
## y_rep[5]     153.69    0.12 5.68   146.41   153.66   160.99  2115    1
## y_rep[6]     157.73    0.13 5.89   150.08   157.84   165.16  2038    1
## y_rep[7]     150.28    0.14 5.78   142.99   150.27   157.62  1710    1
## y_rep[8]     160.28    0.12 5.77   152.93   160.23   167.86  2168    1
## y_rep[9]     145.26    0.13 5.82   137.66   145.14   152.78  2073    1
## y_rep[10]    160.44    0.13 5.69   153.14   160.57   167.72  1806    1
## y_rep[11]    159.60    0.13 5.78   152.26   159.60   166.99  1897    1
## y_rep[12]    153.32    0.14 5.96   145.52   153.34   161.03  1813    1
## y_rep[13]    147.17    0.13 5.76   140.08   146.93   154.52  1992    1
## y_rep[14]    158.55    0.14 5.72   151.15   158.57   165.96  1718    1
## y_rep[15]    143.84    0.14 5.81   136.50   143.88   151.18  1750    1
## y_rep[16]    159.17    0.14 5.71   151.89   159.20   166.68  1757    1
## y_rep[17]    154.68    0.13 5.69   147.25   154.56   161.92  2050    1
## y_rep[18]    150.12    0.13 5.85   142.49   150.16   157.60  1958    1
## y_rep[19]    113.18    0.13 5.72   105.84   113.17   120.34  2000    1
## y_rep[20]     97.15    0.14 5.93    89.53    97.17   104.48  1919    1
## y_rep[21]     85.50    0.13 5.85    77.92    85.67    92.78  1887    1
## y_rep[22]    159.26    0.13 5.77   151.70   159.43   166.55  2035    1
## y_rep[23]    154.97    0.14 5.80   147.01   155.11   162.45  1767    1
## y_rep[24]    123.59    0.14 5.87   115.96   123.49   131.31  1879    1
## y_rep[25]    103.14    0.13 5.75    95.71   102.99   110.73  2022    1
## y_rep[26]    146.28    0.14 5.80   138.99   146.19   153.60  1844    1
## y_rep[27]    149.85    0.13 5.83   142.37   149.95   157.30  1878    1
## y_rep[28]    146.66    0.13 5.66   139.26   146.70   153.97  1956    1
## y_rep[29]    131.94    0.13 5.87   124.54   131.99   139.47  1984    1
## y_rep[30]    121.47    0.13 5.78   114.12   121.41   128.89  1927    1
## y_rep[31]    108.69    0.13 5.89   101.29   108.68   116.06  1925    1
## y_rep[32]    152.51    0.13 5.93   145.19   152.51   160.38  1992    1
## y_rep[33]    160.21    0.13 5.70   152.75   160.16   167.42  1938    1
## y_rep[34]    148.96    0.13 5.68   141.76   148.96   156.12  2016    1
## y_rep[35]    150.61    0.12 5.64   143.44   150.75   157.70  2152    1
## y_rep[36]    157.98    0.13 5.72   150.52   158.05   165.16  2067    1
## y_rep[37]    150.70    0.13 5.86   143.11   150.76   157.99  2028    1
## y_rep[38]    146.62    0.13 5.87   139.28   146.50   154.42  1922    1
## y_rep[39]    158.38    0.14 5.82   150.96   158.34   165.81  1723    1
## y_rep[40]    158.52    0.13 5.78   151.21   158.57   165.86  1888    1
## y_rep[41]    159.41    0.13 5.88   151.95   159.49   166.86  2045    1
## y_rep[42]    159.26    0.13 5.71   151.99   159.34   166.46  1897    1
## y_rep[43]    159.89    0.14 5.72   152.82   159.79   167.33  1691    1
## y_rep[44]    151.40    0.13 6.03   143.64   151.45   159.05  2241    1
## y_rep[45]    159.66    0.13 5.75   152.40   159.61   167.03  1877    1
## y_rep[46]    135.00    0.13 5.80   127.56   135.02   142.29  1965    1
## y_rep[47]    115.55    0.14 5.93   107.96   115.62   123.02  1843    1
## y_rep[48]    122.97    0.13 5.81   115.39   122.96   130.29  1929    1
## y_rep[49]     94.57    0.13 5.79    87.20    94.44   101.93  1978    1
## y_rep[50]    153.91    0.13 5.82   146.32   153.72   161.48  2018    1
## y_rep[51]    150.62    0.13 5.82   143.22   150.85   157.85  1980    1
## y_rep[52]    154.76    0.14 5.92   147.23   154.72   162.43  1898    1
## y_rep[53]    156.83    0.14 5.74   149.86   156.62   164.18  1684    1
## y_rep[54]    119.94    0.13 5.83   112.53   119.95   127.44  1900    1
## y_rep[55]    101.20    0.13 5.79    93.83   101.28   108.77  1906    1
## y_rep[56]     93.16    0.13 5.74    85.82    93.22   100.53  1923    1
## y_rep[57]    159.60    0.13 5.79   152.21   159.55   166.96  2019    1
## y_rep[58]    158.17    0.14 5.87   150.57   158.22   165.45  1836    1
## y_rep[59]    156.12    0.12 5.64   148.86   156.22   163.25  2073    1
## y_rep[60]    160.04    0.13 5.95   152.38   160.14   167.61  2140    1
## y_rep[61]    160.20    0.13 5.92   152.78   160.01   167.94  2086    1
## y_rep[62]    157.49    0.14 5.70   150.06   157.53   164.73  1603    1
## y_rep[63]    158.83    0.14 5.84   151.61   158.60   166.30  1847    1
## y_rep[64]    160.52    0.13 5.74   153.27   160.48   168.20  2065    1
## y_rep[65]    158.49    0.13 5.91   150.78   158.47   166.09  2141    1
## y_rep[66]    157.30    0.12 5.80   149.80   157.25   164.74  2271    1
## y_rep[67]    154.86    0.13 5.80   147.64   154.74   162.41  2048    1
## y_rep[68]    147.86    0.13 5.80   140.43   147.86   155.31  2041    1
## y_rep[69]    160.11    0.13 5.77   152.71   160.18   167.07  2025    1
## y_rep[70]    148.73    0.13 5.80   141.15   148.88   156.25  2059    1
## y_rep[71]    125.79    0.13 5.65   118.51   125.98   132.85  1885    1
## y_rep[72]    128.47    0.13 5.83   120.97   128.39   135.94  1898    1
## y_rep[73]    158.86    0.13 5.89   151.22   158.91   166.26  2005    1
## y_rep[74]    149.10    0.13 5.91   141.72   149.16   156.61  2066    1
## y_rep[75]    136.55    0.13 5.83   129.19   136.38   143.93  1913    1
## y_rep[76]    158.37    0.14 5.93   151.10   158.24   166.03  1787    1
## y_rep[77]    158.35    0.12 5.66   151.41   158.39   165.58  2087    1
## y_rep[78]     93.75    0.13 5.75    86.49    93.70   101.11  2087    1
## y_rep[79]    159.84    0.13 5.68   152.53   159.79   167.37  1901    1
## y_rep[80]    149.86    0.13 5.95   142.32   149.87   157.43  1969    1
## y_rep[81]    154.27    0.13 5.83   146.58   154.34   161.73  1951    1
## y_rep[82]    132.41    0.14 5.78   124.77   132.57   139.44  1760    1
## y_rep[83]    158.38    0.14 5.83   150.71   158.53   165.96  1843    1
## y_rep[84]    155.46    0.13 5.70   147.98   155.45   162.67  1895    1
## y_rep[85]    159.77    0.13 5.83   152.53   159.67   167.35  2000    1
## y_rep[86]    151.39    0.15 5.85   144.14   151.21   158.93  1550    1
## y_rep[87]    107.20    0.13 5.87    99.66   107.29   114.61  1963    1
## y_rep[88]    159.70    0.13 5.83   152.12   159.73   167.21  2069    1
## y_rep[89]    151.59    0.12 5.80   144.17   151.55   159.11  2155    1
## y_rep[90]    147.53    0.14 5.97   139.82   147.38   154.90  1884    1
## y_rep[91]    157.80    0.13 5.67   150.42   157.77   165.23  1793    1
## y_rep[92]    154.82    0.13 5.65   147.70   154.82   161.88  1842    1
## y_rep[93]    158.61    0.13 5.70   151.39   158.55   166.10  1977    1
## y_rep[94]    160.14    0.13 5.83   152.51   160.14   167.58  1907    1
## y_rep[95]    111.07    0.12 5.68   104.04   111.02   118.36  2092    1
## y_rep[96]    158.90    0.14 5.93   151.24   158.88   166.44  1737    1
## y_rep[97]    155.10    0.13 5.92   147.29   155.25   162.79  2105    1
## y_rep[98]    145.73    0.14 5.81   138.45   145.65   153.38  1813    1
## y_rep[99]    108.67    0.13 5.89   101.09   108.71   116.13  1963    1
## y_rep[100]   160.19    0.14 5.83   152.93   160.22   167.79  1735    1
## y_rep[101]   132.22    0.13 5.87   124.54   132.07   139.80  2018    1
## y_rep[102]   159.70    0.13 5.70   152.34   159.84   167.03  1887    1
## y_rep[103]   158.18    0.13 5.83   150.71   158.07   165.57  1936    1
## y_rep[104]   153.21    0.13 5.74   146.04   153.10   160.52  1985    1
## y_rep[105]   141.92    0.13 5.79   134.68   141.82   149.32  1892    1
## y_rep[106]   159.94    0.13 5.71   152.68   159.94   167.43  2026    1
## y_rep[107]   155.26    0.13 5.73   148.11   155.31   162.61  2033    1
## y_rep[108]   152.24    0.13 5.91   144.53   152.21   159.84  2050    1
## y_rep[109]   141.75    0.13 5.82   134.30   141.66   149.33  1896    1
## y_rep[110]   106.48    0.13 5.82    98.85   106.39   113.97  2026    1
## y_rep[111]   104.81    0.13 5.90    97.23   104.77   112.41  1926    1
## y_rep[112]   157.42    0.13 5.87   149.67   157.44   164.88  2008    1
## y_rep[113]   153.55    0.13 5.68   146.52   153.44   161.04  1901    1
## y_rep[114]   109.89    0.13 5.62   102.62   109.99   117.14  1925    1
## y_rep[115]   100.25    0.14 5.79    92.59   100.38   107.50  1658    1
## y_rep[116]    88.76    0.13 5.76    81.12    88.57    95.95  1867    1
## y_rep[117]   158.50    0.12 5.73   151.22   158.53   165.96  2101    1
## y_rep[118]   149.17    0.13 5.80   141.77   149.21   156.72  1999    1
## y_rep[119]   156.90    0.15 5.78   149.35   156.93   164.13  1516    1
## y_rep[120]   154.59    0.13 5.77   147.34   154.44   162.06  1901    1
## y_rep[121]   160.44    0.13 5.80   153.07   160.41   167.68  2050    1
## y_rep[122]   160.52    0.13 5.75   153.05   160.49   167.79  2094    1
## y_rep[123]   160.30    0.13 5.81   152.70   160.32   167.49  1885    1
## y_rep[124]    83.23    0.13 5.92    75.72    83.22    90.92  2067    1
## y_rep[125]   154.77    0.13 5.95   147.19   154.87   162.42  2006    1
## y_rep[126]   159.41    0.13 5.86   151.80   159.47   167.07  2101    1
## y_rep[127]    94.17    0.14 5.89    86.70    94.23   101.76  1781    1
## y_rep[128]   154.28    0.13 5.78   146.98   154.07   161.76  1891    1
## y_rep[129]   160.31    0.13 5.82   153.01   160.28   167.89  2013    1
## y_rep[130]   154.34    0.13 5.74   147.15   154.27   161.91  1921    1
## y_rep[131]   112.80    0.13 5.80   105.39   112.85   120.33  1856    1
## y_rep[132]   100.34    0.12 5.63    92.93   100.38   107.78  2147    1
## y_rep[133]   160.38    0.13 5.86   152.86   160.32   168.21  2009    1
## y_rep[134]   159.67    0.14 5.96   152.08   159.81   167.31  1940    1
## y_rep[135]   154.97    0.14 5.71   147.70   155.11   162.18  1711    1
## y_rep[136]   159.29    0.13 5.80   151.85   159.27   166.77  1993    1
## y_rep[137]   150.62    0.14 5.81   143.28   150.57   158.15  1642    1
## y_rep[138]   159.51    0.13 5.75   152.02   159.66   166.76  1905    1
## y_rep[139]   135.91    0.12 5.60   128.83   135.88   143.13  2027    1
## y_rep[140]   159.20    0.13 5.78   151.44   159.37   166.37  1978    1
## y_rep[141]    89.69    0.14 5.96    81.86    89.60    97.44  1887    1
## y_rep[142]   158.51    0.14 5.91   151.00   158.45   166.23  1707    1
## y_rep[143]   151.50    0.13 5.76   144.22   151.46   158.97  1897    1
## y_rep[144]   150.12    0.13 5.69   142.79   150.20   157.37  1921    1
## y_rep[145]   119.78    0.14 5.88   112.13   119.77   127.34  1852    1
## y_rep[146]   148.43    0.13 5.90   140.83   148.34   156.06  1939    1
## y_rep[147]   154.44    0.13 5.78   146.91   154.52   161.63  2020    1
## y_rep[148]   113.89    0.13 5.72   106.56   113.74   121.23  1851    1
## y_rep[149]   105.56    0.14 6.01    97.87   105.28   113.32  1957    1
## y_rep[150]   156.73    0.13 5.93   148.80   156.79   164.38  2081    1
## y_rep[151]   158.60    0.13 5.68   151.36   158.78   165.74  1960    1
## y_rep[152]   117.36    0.13 5.77   109.95   117.36   124.86  1913    1
## y_rep[153]   111.41    0.14 5.92   103.90   111.30   119.14  1885    1
## y_rep[154]   160.41    0.13 5.65   153.18   160.50   167.77  1783    1
## y_rep[155]   142.72    0.14 5.73   135.49   142.83   149.89  1791    1
## y_rep[156]   153.32    0.13 5.73   145.86   153.22   160.98  1979    1
## y_rep[157]    94.97    0.13 5.88    87.40    94.95   102.27  1976    1
## y_rep[158]   159.83    0.12 5.80   152.47   159.89   167.23  2181    1
## y_rep[159]   159.29    0.13 5.90   151.99   159.27   166.72  2035    1
## y_rep[160]   160.48    0.13 5.88   152.90   160.46   167.97  2020    1
## y_rep[161]   160.34    0.13 5.73   152.84   160.46   167.48  2013    1
## y_rep[162]   160.06    0.13 5.85   152.48   160.25   167.56  2042    1
## y_rep[163]   154.70    0.13 5.94   147.17   154.81   162.22  1972    1
## y_rep[164]   120.72    0.13 5.74   113.24   120.61   128.17  1868    1
## y_rep[165]   153.64    0.14 5.81   146.37   153.64   161.15  1787    1
## y_rep[166]   159.33    0.13 5.63   152.15   159.36   166.53  1897    1
## y_rep[167]   156.32    0.14 5.87   148.90   156.31   163.79  1765    1
## y_rep[168]   157.77    0.12 5.78   150.48   157.68   165.35  2167    1
## y_rep[169]   154.42    0.13 5.90   146.87   154.45   161.81  2067    1
## y_rep[170]   159.99    0.13 5.69   152.78   159.78   167.14  1956    1
## y_rep[171]    93.32    0.13 5.69    86.11    93.29   100.55  1919    1
## y_rep[172]    94.23    0.13 5.66    86.78    94.16   101.39  1961    1
## y_rep[173]   156.21    0.14 5.91   148.59   156.04   164.03  1869    1
## y_rep[174]   141.26    0.13 5.76   133.99   141.29   148.62  2098    1
## y_rep[175]   159.85    0.13 5.97   152.06   160.03   167.46  2014    1
## y_rep[176]   160.23    0.13 5.72   152.74   160.26   167.58  1990    1
## y_rep[177]   147.99    0.13 5.66   140.74   147.92   155.40  2018    1
## y_rep[178]   159.06    0.13 5.57   152.15   158.99   166.21  1965    1
## y_rep[179]   159.85    0.13 5.74   152.56   159.78   167.20  1937    1
## y_rep[180]   154.49    0.13 5.80   146.93   154.56   161.76  2025    1
## y_rep[181]   159.59    0.15 5.89   152.07   159.48   167.07  1546    1
## y_rep[182]   157.96    0.14 5.98   150.16   157.94   165.93  1874    1
## y_rep[183]   158.32    0.13 5.79   151.02   158.32   165.77  1978    1
## y_rep[184]   158.99    0.13 5.79   151.62   159.19   166.13  1918    1
## y_rep[185]   140.16    0.13 5.86   132.60   140.19   147.80  2140    1
## y_rep[186]   141.22    0.13 5.88   133.67   141.37   148.59  1944    1
## y_rep[187]   123.91    0.13 5.79   116.48   123.92   131.12  1876    1
## y_rep[188]   125.47    0.13 5.90   117.79   125.40   133.14  1989    1
## y_rep[189]   118.66    0.13 5.65   111.50   118.80   125.78  1980    1
## y_rep[190]   146.24    0.13 5.84   138.94   146.21   153.75  2095    1
## y_rep[191]   157.37    0.13 5.74   150.12   157.35   164.57  1920    1
## y_rep[192]   156.07    0.13 5.87   148.73   156.07   163.79  1950    1
## y_rep[193]   157.40    0.13 5.81   150.02   157.49   164.74  2007    1
## y_rep[194]   151.13    0.13 5.84   143.55   151.29   158.67  2102    1
## y_rep[195]   153.65    0.12 5.62   146.73   153.50   160.92  2090    1
## y_rep[196]    93.88    0.13 5.94    86.35    93.79   101.49  1970    1
## y_rep[197]   157.47    0.13 5.91   150.03   157.41   165.38  1962    1
## y_rep[198]   160.41    0.14 5.84   152.91   160.44   167.83  1845    1
## y_rep[199]   160.32    0.13 5.67   153.02   160.34   167.69  1895    1
## y_rep[200]   109.74    0.13 5.73   102.30   109.86   116.96  1912    1
## y_rep[201]    96.42    0.14 5.91    89.13    96.39   103.98  1897    1
## y_rep[202]   151.46    0.13 5.71   144.16   151.34   158.89  2024    1
## y_rep[203]    92.47    0.13 5.80    84.92    92.56    99.87  2042    1
## y_rep[204]   157.26    0.13 5.74   150.13   157.31   164.48  2023    1
## y_rep[205]   152.67    0.14 5.85   145.26   152.73   160.05  1824    1
## y_rep[206]   160.14    0.14 5.96   152.58   160.07   167.96  1889    1
## y_rep[207]   150.89    0.12 5.70   143.58   150.85   158.44  2086    1
## y_rep[208]   151.43    0.13 5.79   143.90   151.41   158.87  1865    1
## y_rep[209]   156.46    0.13 5.76   149.21   156.41   163.97  1942    1
## y_rep[210]   151.87    0.14 6.01   144.30   151.72   160.02  1729    1
## y_rep[211]   139.15    0.13 5.90   131.55   139.15   146.59  1993    1
## y_rep[212]   157.95    0.13 5.89   150.47   157.91   165.43  2056    1
## y_rep[213]   120.74    0.13 5.95   113.13   120.76   128.34  1975    1
## y_rep[214]   153.08    0.13 5.78   145.78   153.13   160.58  2024    1
## y_rep[215]   159.52    0.13 5.85   151.77   159.46   166.87  2091    1
## y_rep[216]   153.56    0.13 5.90   146.00   153.51   161.24  2010    1
## y_rep[217]   107.76    0.13 5.72   100.31   107.79   115.05  2029    1
## y_rep[218]   160.31    0.13 5.94   152.64   160.23   168.26  2104    1
## y_rep[219]   157.07    0.13 5.74   149.62   156.96   164.44  1853    1
## y_rep[220]   156.83    0.13 5.71   149.40   156.81   164.12  1866    1
## y_rep[221]    98.48    0.13 5.72    90.97    98.37   106.00  2013    1
## y_rep[222]   112.16    0.13 5.81   104.78   112.06   119.69  1917    1
## y_rep[223]   146.95    0.15 5.78   139.57   147.04   154.28  1460    1
## y_rep[224]   156.92    0.13 5.78   149.63   156.82   164.34  2011    1
## y_rep[225]   158.84    0.13 5.75   151.37   158.90   166.21  1929    1
## y_rep[226]   157.45    0.13 5.77   150.22   157.49   164.67  1837    1
## y_rep[227]   116.74    0.13 5.79   109.23   116.81   124.15  2094    1
## y_rep[228]   159.04    0.13 5.76   151.61   158.94   166.46  1843    1
## y_rep[229]   153.50    0.14 5.78   146.29   153.45   160.89  1778    1
## y_rep[230]   150.35    0.14 5.85   142.98   150.38   157.83  1855    1
## y_rep[231]   154.42    0.14 5.72   147.06   154.55   161.80  1682    1
## y_rep[232]   108.22    0.13 5.82   100.85   108.27   115.56  1950    1
## y_rep[233]   150.02    0.13 5.72   142.65   150.04   157.42  2051    1
## y_rep[234]   159.06    0.13 5.59   151.96   159.03   166.04  1954    1
## y_rep[235]   151.06    0.14 5.71   143.91   151.02   158.35  1718    1
## y_rep[236]   151.93    0.13 5.65   144.76   151.99   159.30  1800    1
## y_rep[237]   159.71    0.13 5.82   152.27   159.71   167.14  1945    1
## y_rep[238]   160.53    0.13 5.94   152.98   160.66   168.04  1945    1
## y_rep[239]   157.26    0.13 5.75   149.82   157.33   164.56  1922    1
## y_rep[240]   153.16    0.13 5.81   145.85   153.20   160.62  1998    1
## y_rep[241]   131.56    0.13 5.74   124.15   131.69   138.67  2045    1
## y_rep[242]   151.66    0.12 5.63   144.69   151.68   158.65  2059    1
## y_rep[243]   115.87    0.12 5.72   108.27   115.84   123.06  2230    1
## y_rep[244]   155.32    0.14 5.83   147.64   155.57   162.49  1774    1
## y_rep[245]   140.68    0.14 5.68   133.38   140.75   147.89  1737    1
## y_rep[246]   116.99    0.13 5.88   109.52   116.96   124.59  1941    1
## y_rep[247]   157.29    0.13 5.76   149.86   157.31   164.77  2049    1
## y_rep[248]   149.80    0.13 5.68   142.47   149.74   157.03  1999    1
## y_rep[249]   103.05    0.14 5.96    95.40   103.06   110.84  1763    1
## y_rep[250]   147.33    0.13 5.62   140.31   147.31   154.80  1899    1
## y_rep[251]   153.11    0.13 5.85   145.52   153.16   160.67  2031    1
## y_rep[252]   158.59    0.13 5.70   151.33   158.71   165.90  1909    1
## y_rep[253]   158.43    0.13 5.72   151.00   158.43   165.80  2031    1
## y_rep[254]   157.86    0.14 5.85   150.42   157.74   165.37  1866    1
## y_rep[255]   153.44    0.13 5.91   145.89   153.33   161.02  2058    1
## y_rep[256]   147.90    0.13 5.84   140.48   148.07   155.17  1994    1
## y_rep[257]   155.42    0.13 5.76   147.83   155.53   162.74  2075    1
## y_rep[258]   158.95    0.13 5.73   151.73   159.04   166.06  2055    1
## y_rep[259]   148.14    0.13 5.82   140.86   148.10   155.63  1967    1
## y_rep[260]   160.31    0.13 5.83   152.98   160.19   167.78  2008    1
## y_rep[261]   150.71    0.13 5.65   143.61   150.55   157.81  1975    1
## y_rep[262]   158.34    0.13 5.94   150.74   158.48   165.99  2066    1
## y_rep[263]   147.52    0.14 5.71   140.03   147.48   154.86  1687    1
## y_rep[264]    97.91    0.13 5.69    90.51    98.06   104.94  2013    1
## y_rep[265]   153.60    0.13 5.85   146.16   153.52   161.15  2009    1
## y_rep[266]   100.80    0.12 5.67    93.52   100.81   107.87  2070    1
## y_rep[267]   156.47    0.13 5.79   149.00   156.40   164.02  2051    1
## y_rep[268]   159.70    0.13 5.83   152.19   159.70   167.32  1880    1
## y_rep[269]   160.24    0.13 5.80   152.95   160.11   167.85  2013    1
## y_rep[270]   149.87    0.13 5.69   142.62   149.85   157.16  1983    1
## y_rep[271]   146.67    0.13 5.76   139.20   146.57   154.16  1928    1
## y_rep[272]   159.63    0.15 5.82   152.20   159.85   166.80  1477    1
## y_rep[273]   118.34    0.14 5.92   110.77   118.31   126.06  1825    1
## y_rep[274]   156.39    0.13 5.63   149.22   156.42   163.61  1909    1
## y_rep[275]   156.49    0.14 5.77   149.29   156.45   163.88  1787    1
## y_rep[276]   156.58    0.13 5.76   149.41   156.48   163.88  1917    1
## y_rep[277]   156.61    0.13 5.85   149.28   156.65   164.04  2025    1
## y_rep[278]   152.57    0.13 5.79   145.08   152.46   160.05  1893    1
## y_rep[279]   156.03    0.13 5.77   148.60   156.14   163.37  2083    1
## y_rep[280]   125.59    0.13 5.72   118.31   125.69   132.82  2037    1
## y_rep[281]   144.81    0.14 5.89   137.39   144.73   152.59  1780    1
## y_rep[282]   142.05    0.13 5.75   134.62   142.17   149.32  2079    1
## y_rep[283]   147.18    0.13 5.80   139.75   147.23   154.62  1975    1
## y_rep[284]   159.57    0.13 5.72   152.18   159.61   167.07  2003    1
## y_rep[285]   160.18    0.13 5.85   152.73   160.13   167.54  1941    1
## y_rep[286]   148.79    0.13 5.79   141.59   148.71   156.23  1975    1
## y_rep[287]   152.99    0.13 5.70   145.80   152.83   160.42  2055    1
## y_rep[288]   159.61    0.13 5.77   152.17   159.60   166.93  1873    1
## y_rep[289]   156.14    0.13 5.73   148.79   156.14   163.30  1934    1
## y_rep[290]   144.00    0.13 5.85   136.44   144.02   151.31  2066    1
## y_rep[291]   118.24    0.13 5.93   110.74   118.05   126.03  2016    1
## y_rep[292]   105.20    0.12 5.80    97.77   105.00   112.76  2226    1
## y_rep[293]   159.45    0.13 5.87   151.78   159.39   167.26  1930    1
## y_rep[294]   143.44    0.13 5.89   135.92   143.52   150.79  2041    1
## y_rep[295]   156.05    0.12 5.70   148.71   155.84   163.52  2092    1
## y_rep[296]   158.62    0.14 5.79   151.35   158.56   166.07  1779    1
## y_rep[297]   155.19    0.13 5.69   148.08   155.08   162.32  2010    1
## y_rep[298]   145.43    0.13 5.94   137.80   145.29   153.29  2076    1
## y_rep[299]   124.81    0.13 5.79   117.62   124.58   132.30  2040    1
## y_rep[300]   144.12    0.13 5.80   136.61   144.09   151.41  2005    1
## y_rep[301]    90.80    0.14 5.99    83.31    90.71    98.39  1875    1
## y_rep[302]   153.67    0.12 5.66   146.41   153.76   160.81  2083    1
## y_rep[303]   152.22    0.13 5.89   144.64   152.20   159.80  2041    1
## y_rep[304]   147.17    0.13 5.84   139.56   147.27   154.45  2053    1
## y_rep[305]   160.07    0.13 5.79   152.61   160.00   167.37  1844    1
## y_rep[306]   157.24    0.13 5.80   149.91   157.14   164.83  2154    1
## y_rep[307]   127.57    0.13 5.66   120.37   127.48   134.78  1779    1
## y_rep[308]   101.38    0.13 5.78    93.68   101.33   108.96  2001    1
## y_rep[309]   159.20    0.13 5.72   152.04   159.14   166.59  2062    1
## y_rep[310]   159.49    0.14 5.93   151.85   159.48   166.88  1764    1
## y_rep[311]   155.56    0.14 5.85   148.06   155.55   163.22  1678    1
## y_rep[312]   108.60    0.13 5.82   101.24   108.63   115.87  1978    1
## y_rep[313]   160.16    0.13 5.76   153.11   160.07   167.57  2011    1
## y_rep[314]   159.88    0.13 5.78   152.58   159.84   167.18  2113    1
## y_rep[315]   144.44    0.14 5.69   137.13   144.56   151.74  1669    1
## y_rep[316]   151.67    0.13 5.74   144.45   151.50   158.93  1932    1
## y_rep[317]   159.33    0.14 5.84   152.01   159.27   166.96  1844    1
## y_rep[318]   152.73    0.14 6.03   145.18   152.61   160.54  1793    1
## y_rep[319]   160.01    0.14 5.77   152.87   160.05   167.50  1700    1
## y_rep[320]   152.33    0.13 5.78   145.09   152.34   159.75  1895    1
## y_rep[321]   155.99    0.13 5.82   148.33   156.21   163.40  2130    1
## y_rep[322]   160.35    0.13 5.74   152.80   160.52   167.76  2088    1
## y_rep[323]   156.68    0.14 5.88   149.24   156.80   164.16  1876    1
## y_rep[324]    88.85    0.13 5.70    81.51    88.99    95.90  2024    1
## y_rep[325]    89.91    0.15 5.62    82.63    89.93    97.14  1399    1
## y_rep[326]    69.14    0.13 5.61    62.10    69.16    76.19  1958    1
## y_rep[327]    90.98    0.13 5.75    83.70    91.00    98.34  1992    1
## y_rep[328]   110.02    0.13 5.68   102.74   110.12   117.39  1988    1
## y_rep[329]    88.64    0.13 5.86    81.14    88.74    96.03  1973    1
## y_rep[330]   153.69    0.12 5.63   146.45   153.73   160.90  2126    1
## y_rep[331]   103.94    0.14 5.84    96.46   104.05   111.24  1731    1
## y_rep[332]    72.55    0.13 5.74    65.19    72.50    79.99  2036    1
## y_rep[333]    90.17    0.13 5.84    82.56    90.28    97.58  1898    1
## y_rep[334]   100.25    0.13 5.67    92.94   100.32   107.34  1874    1
## y_rep[335]    83.38    0.13 5.97    75.89    83.32    91.10  2005    1
## y_rep[336]   154.37    0.13 5.82   146.99   154.30   161.84  1939    1
## y_rep[337]   155.05    0.13 5.86   147.54   154.97   162.33  1974    1
## y_rep[338]   135.18    0.13 5.87   127.43   135.17   143.07  1988    1
## y_rep[339]    93.11    0.13 5.78    85.66    93.24   100.39  1906    1
## y_rep[340]   155.11    0.13 5.86   147.51   155.24   162.55  1990    1
## y_rep[341]   153.17    0.14 5.79   145.83   153.22   160.38  1802    1
## y_rep[342]   155.02    0.13 5.80   147.42   155.11   162.46  1991    1
## y_rep[343]   157.74    0.13 5.84   150.12   157.70   165.61  1985    1
## y_rep[344]   116.10    0.13 5.79   108.51   116.07   123.62  2010    1
## y_rep[345]   102.84    0.13 5.83    95.39   102.70   110.47  1970    1
## y_rep[346]   158.92    0.14 5.92   151.34   159.03   166.35  1814    1
## y_rep[347]   155.02    0.12 5.71   147.68   154.98   162.47  2132    1
## y_rep[348]   106.17    0.13 5.82    98.82   106.06   113.86  2116    1
## y_rep[349]    94.69    0.13 5.77    87.19    94.76   102.35  2106    1
## y_rep[350]   159.02    0.13 5.88   151.37   158.98   166.57  2035    1
## y_rep[351]   157.60    0.13 5.79   150.29   157.44   165.13  1980    1
## y_rep[352]   160.21    0.13 5.75   152.88   160.20   167.60  1953    1
## y_rep[353]   158.39    0.13 5.72   151.19   158.26   165.93  1904    1
## y_rep[354]    93.48    0.13 5.76    86.16    93.39   100.99  1972    1
## y_rep[355]    68.28    0.14 6.00    60.41    68.30    75.92  1873    1
## y_rep[356]   135.15    0.13 5.85   127.64   135.13   142.71  2024    1
## y_rep[357]   155.86    0.13 5.74   148.52   155.78   163.29  1893    1
## y_rep[358]   155.88    0.13 5.83   148.28   155.90   163.28  1963    1
## y_rep[359]    88.79    0.13 5.77    81.53    88.85    96.21  2024    1
## y_rep[360]    89.62    0.13 5.87    82.16    89.57    97.37  1998    1
## y_rep[361]    75.32    0.14 5.91    67.69    75.49    82.72  1860    1
## y_rep[362]    92.82    0.13 5.65    85.60    92.91   100.01  1987    1
## y_rep[363]    74.05    0.14 5.71    66.87    73.96    81.17  1553    1
## y_rep[364]    91.97    0.13 5.82    84.47    91.85    99.43  1943    1
## y_rep[365]    76.03    0.14 6.03    68.12    76.24    83.58  1833    1
## y_rep[366]   160.35    0.13 5.93   153.00   160.20   168.17  2014    1
## y_rep[367]   156.70    0.13 5.76   149.33   156.66   164.20  1976    1
## y_rep[368]    92.28    0.13 5.75    84.72    92.47    99.60  2105    1
## y_rep[369]    92.81    0.13 5.84    85.43    92.82   100.33  2057    1
## y_rep[370]    74.67    0.13 5.84    67.23    74.66    82.08  1989    1
## y_rep[371]    81.05    0.13 5.85    73.51    81.18    88.52  1964    1
## y_rep[372]   158.10    0.14 5.82   150.35   157.98   165.62  1837    1
## y_rep[373]   134.57    0.13 5.73   127.01   134.56   141.93  1934    1
## y_rep[374]   140.36    0.14 5.68   133.15   140.40   147.63  1728    1
## y_rep[375]   160.22    0.14 5.81   152.87   160.12   167.78  1622    1
## y_rep[376]   114.34    0.13 5.85   106.83   114.13   121.97  1884    1
## y_rep[377]    78.55    0.14 5.77    71.13    78.47    85.81  1799    1
## y_rep[378]    78.13    0.13 5.67    70.92    78.15    85.32  1849    1
## y_rep[379]    89.33    0.14 5.82    81.84    89.43    96.84  1667    1
## y_rep[380]   160.24    0.13 5.90   152.56   160.15   168.13  2098    1
## y_rep[381]    76.56    0.13 5.86    69.09    76.49    84.08  2077    1
## y_rep[382]   131.83    0.13 5.73   124.29   131.99   139.39  2028    1
## y_rep[383]   158.39    0.13 5.74   150.90   158.23   165.81  1829    1
## y_rep[384]    86.71    0.14 5.79    79.37    86.67    94.06  1775    1
## y_rep[385]    97.05    0.13 5.70    89.66    97.04   104.37  1942    1
## y_rep[386]   156.90    0.13 5.74   149.54   156.94   164.15  1945    1
## y_rep[387]   157.34    0.13 5.92   149.80   157.14   164.98  1954    1
## y_rep[388]   159.13    0.13 5.68   151.92   158.99   166.45  2012    1
## y_rep[389]   157.90    0.14 5.95   150.56   157.92   165.62  1888    1
## y_rep[390]   114.99    0.13 5.88   107.53   114.79   122.70  2042    1
## y_rep[391]   109.45    0.13 5.78   101.98   109.46   116.82  1924    1
## y_rep[392]    87.01    0.13 5.79    79.35    87.10    94.50  1990    1
## y_rep[393]   156.02    0.14 5.62   148.92   155.96   163.06  1730    1
## y_rep[394]   131.63    0.13 5.82   124.42   131.41   139.13  1954    1
## y_rep[395]   111.90    0.13 5.93   104.14   111.96   119.40  1991    1
## y_rep[396]    96.69    0.13 5.69    89.47    96.69   104.15  1870    1
## y_rep[397]   160.23    0.13 5.78   152.91   160.18   167.61  1903    1
## y_rep[398]   152.57    0.12 5.70   145.20   152.61   159.86  2087    1
## y_rep[399]   130.14    0.14 5.78   122.59   130.16   137.71  1755    1
## y_rep[400]   159.87    0.13 5.75   152.57   159.75   167.38  1855    1
## y_rep[401]    81.16    0.14 5.77    73.76    81.08    88.44  1801    1
## y_rep[402]    75.23    0.13 5.76    68.26    75.23    82.72  1891    1
## y_rep[403]   157.84    0.14 5.74   150.51   157.76   165.10  1764    1
## y_rep[404]   112.72    0.12 5.63   105.48   112.69   120.03  2236    1
## y_rep[405]   149.88    0.13 5.64   142.51   149.91   157.39  1956    1
## y_rep[406]   159.25    0.13 5.65   151.88   159.35   166.32  1825    1
## y_rep[407]    74.26    0.13 5.90    66.57    74.20    81.81  1926    1
## y_rep[408]   133.48    0.12 5.67   126.23   133.48   140.61  2166    1
## y_rep[409]    76.15    0.13 5.90    68.69    76.09    83.86  2021    1
## y_rep[410]   156.44    0.13 5.80   148.86   156.47   163.81  1930    1
## y_rep[411]   159.86    0.15 5.82   152.50   159.78   167.24  1477    1
## y_rep[412]   153.11    0.13 5.68   145.48   153.17   160.21  2010    1
## y_rep[413]   149.19    0.13 5.70   141.92   149.13   156.49  1909    1
## y_rep[414]   160.26    0.13 5.90   152.80   160.26   167.77  2049    1
## y_rep[415]   156.66    0.13 5.75   149.42   156.81   163.89  1892    1
## y_rep[416]    76.84    0.13 5.69    69.55    76.72    84.32  2013    1
## y_rep[417]   155.42    0.14 5.81   148.03   155.41   162.90  1724    1
## y_rep[418]   158.08    0.13 5.90   150.49   158.04   165.63  2018    1
## y_rep[419]   151.61    0.13 5.73   144.27   151.61   159.17  1864    1
## y_rep[420]   145.46    0.13 5.80   137.78   145.52   152.93  1979    1
## y_rep[421]   121.72    0.13 5.89   114.32   121.46   129.44  2151    1
## y_rep[422]   151.13    0.14 5.78   143.94   151.13   158.66  1801    1
## y_rep[423]   149.24    0.13 5.91   141.60   149.23   156.75  2122    1
## y_rep[424]   159.01    0.13 5.89   151.91   158.81   166.61  2020    1
## y_rep[425]   146.59    0.13 5.76   139.30   146.60   153.96  1858    1
## y_rep[426]    81.73    0.14 5.90    74.25    81.80    89.36  1816    1
## y_rep[427]   156.08    0.13 5.76   148.71   156.16   163.61  1989    1
## y_rep[428]   158.92    0.14 5.82   151.25   158.89   166.33  1799    1
## y_rep[429]   136.87    0.13 5.84   129.35   137.01   144.17  1931    1
## y_rep[430]   160.03    0.14 5.80   152.59   159.92   167.40  1831    1
## y_rep[431]   156.17    0.13 5.66   149.14   156.18   163.20  1844    1
## y_rep[432]   133.79    0.13 5.89   126.30   133.72   141.26  2064    1
## y_rep[433]   115.32    0.13 5.96   107.69   115.41   122.99  2163    1
## y_rep[434]   158.56    0.13 5.79   151.08   158.54   166.01  2042    1
## y_rep[435]   151.45    0.13 5.85   144.07   151.36   159.03  1977    1
## y_rep[436]   158.00    0.13 5.66   150.84   157.91   165.45  1849    1
## y_rep[437]   160.05    0.13 5.80   152.80   160.15   167.49  1949    1
## y_rep[438]   158.93    0.13 5.71   151.36   158.94   166.30  1984    1
## y_rep[439]   159.01    0.13 5.87   151.71   158.84   166.55  1898    1
## y_rep[440]    94.94    0.13 5.76    87.69    94.85   102.37  2070    1
## y_rep[441]    71.81    0.12 5.74    64.54    71.86    79.16  2170    1
## y_rep[442]   158.42    0.13 5.73   151.16   158.44   165.75  2067    1
## y_rep[443]   146.57    0.13 5.91   139.04   146.53   154.22  2205    1
## y_rep[444]   148.05    0.12 5.69   140.88   147.96   155.47  2135    1
## y_rep[445]   159.18    0.14 5.72   151.88   159.18   166.52  1753    1
## y_rep[446]   160.12    0.13 5.86   152.62   160.08   167.53  2021    1
## y_rep[447]   158.20    0.13 5.87   150.56   158.19   165.85  1990    1
## y_rep[448]   148.56    0.13 5.69   141.02   148.58   155.93  1835    1
## y_rep[449]   126.87    0.14 5.75   119.39   126.99   134.32  1805    1
## y_rep[450]   103.95    0.13 5.79    96.43   103.98   111.14  2015    1
## y_rep[451]   159.22    0.13 5.77   151.71   159.18   166.56  2069    1
## y_rep[452]   150.10    0.13 5.74   142.72   150.06   157.52  2087    1
## y_rep[453]    78.69    0.13 5.92    70.95    78.54    86.42  2003    1
## y_rep[454]   158.17    0.13 5.76   150.82   158.24   165.41  1855    1
## y_rep[455]   133.97    0.13 5.72   126.57   133.99   141.43  2015    1
## y_rep[456]    77.44    0.13 5.74    70.13    77.48    84.84  1971    1
## y_rep[457]    73.98    0.13 5.84    66.76    74.05    81.48  1997    1
## y_rep[458]   160.54    0.13 5.91   152.74   160.64   167.93  2023    1
## y_rep[459]   141.61    0.13 5.77   134.17   141.57   148.97  2071    1
## y_rep[460]   160.34    0.14 6.02   152.70   160.31   167.88  1925    1
## y_rep[461]   159.15    0.12 5.70   151.72   158.97   166.61  2079    1
## y_rep[462]   155.17    0.14 5.59   147.93   155.32   162.13  1690    1
## y_rep[463]    74.15    0.13 5.89    66.96    74.14    81.82  1991    1
## y_rep[464]   144.75    0.13 5.83   137.13   144.59   152.06  1954    1
## y_rep[465]   159.85    0.13 5.78   152.36   159.89   167.35  1980    1
## y_rep[466]   153.70    0.13 5.80   146.26   153.73   160.92  2041    1
## y_rep[467]    65.02    0.14 5.92    57.51    64.90    72.67  1858    1
## y_rep[468]    70.59    0.14 5.89    63.23    70.59    78.11  1729    1
## y_rep[469]   156.35    0.12 5.80   148.94   156.32   163.78  2225    1
## y_rep[470]   143.53    0.13 5.75   136.12   143.64   150.65  1953    1
## y_rep[471]   106.05    0.13 5.69    98.82   105.97   113.20  1943    1
## y_rep[472]    77.56    0.13 5.94    69.92    77.52    85.24  2248    1
## y_rep[473]   158.44    0.13 5.62   151.44   158.40   165.73  1792    1
## y_rep[474]   157.68    0.13 5.70   150.49   157.82   164.90  1828    1
## y_rep[475]   114.91    0.13 5.79   107.39   115.01   122.09  1877    1
## y_rep[476]    92.94    0.13 5.84    85.33    93.14   100.44  1882    1
## y_rep[477]   158.40    0.13 5.74   151.20   158.32   165.73  1930    1
## y_rep[478]   153.18    0.13 5.69   145.68   153.20   160.57  2064    1
## y_rep[479]   159.72    0.13 5.85   152.13   159.81   167.19  1883    1
## y_rep[480]   156.95    0.13 5.78   149.41   157.17   164.30  2015    1
## y_rep[481]    89.53    0.13 5.72    82.35    89.45    96.70  1904    1
## y_rep[482]   154.44    0.13 5.81   146.93   154.58   161.91  2030    1
## y_rep[483]   154.05    0.12 5.63   146.93   154.02   161.32  2063    1
## y_rep[484]    81.09    0.13 5.61    73.86    81.12    88.21  1745    1
## y_rep[485]   156.94    0.13 5.75   149.42   156.94   164.30  2074    1
## y_rep[486]   160.02    0.13 5.75   153.04   159.79   167.62  2084    1
## y_rep[487]   117.67    0.13 5.70   110.36   117.55   124.89  1927    1
## y_rep[488]    89.18    0.13 5.92    81.45    89.17    96.76  2036    1
## y_rep[489]   159.39    0.13 5.76   151.89   159.54   166.70  2029    1
## y_rep[490]   133.39    0.13 5.79   125.92   133.48   140.78  1917    1
## y_rep[491]   160.06    0.14 5.90   152.51   160.01   167.57  1748    1
## y_rep[492]   155.99    0.13 5.86   148.22   156.12   163.54  1915    1
## y_rep[493]   108.06    0.13 5.79   100.61   108.02   115.60  1842    1
## y_rep[494]   146.26    0.13 5.81   138.75   146.21   153.91  2086    1
## y_rep[495]   144.56    0.14 5.83   137.25   144.35   152.46  1825    1
## y_rep[496]   155.10    0.13 5.73   148.06   154.84   162.58  2059    1
## y_rep[497]   141.85    0.13 5.83   134.53   141.82   149.46  2041    1
## y_rep[498]   158.07    0.13 5.80   150.37   158.09   165.33  2036    1
## y_rep[499]   154.18    0.13 5.78   146.76   154.15   161.61  2039    1
## y_rep[500]   149.99    0.13 5.75   142.57   149.92   157.51  1964    1
## y_rep[501]   155.60    0.13 5.81   148.07   155.68   163.13  2094    1
## y_rep[502]   160.43    0.13 5.85   152.67   160.46   167.89  2160    1
## y_rep[503]   155.46    0.13 5.75   148.21   155.39   162.78  2032    1
## y_rep[504]   118.44    0.13 5.91   110.96   118.48   126.05  1938    1
## y_rep[505]    88.81    0.13 5.79    81.53    88.84    96.38  1954    1
## y_rep[506]   152.15    0.13 5.75   145.01   152.14   159.35  2054    1
## y_rep[507]   155.58    0.12 5.61   148.23   155.49   162.77  2212    1
## y_rep[508]   101.68    0.13 5.81    94.24   101.76   108.82  2037    1
## y_rep[509]   119.50    0.14 5.80   111.94   119.68   126.66  1765    1
## y_rep[510]   160.24    0.13 5.74   152.76   160.41   167.57  1870    1
## y_rep[511]   150.51    0.14 5.72   142.96   150.50   158.02  1794    1
## y_rep[512]   112.59    0.14 5.83   105.33   112.48   119.89  1774    1
## y_rep[513]    87.53    0.13 5.71    80.27    87.53    94.69  1840    1
## y_rep[514]    95.51    0.13 5.74    88.06    95.61   102.72  1984    1
## y_rep[515]   155.28    0.12 5.75   147.92   155.19   162.78  2172    1
## y_rep[516]   147.15    0.12 5.68   139.74   147.24   154.21  2162    1
## y_rep[517]   156.18    0.12 5.68   149.09   156.18   163.32  2134    1
## y_rep[518]    62.99    0.13 5.82    55.64    62.87    70.50  2062    1
## y_rep[519]   149.95    0.13 5.74   142.69   149.83   157.47  2014    1
## y_rep[520]    66.28    0.12 5.61    59.07    66.25    73.38  2124    1
## y_rep[521]    97.97    0.13 5.96    90.37    97.95   105.60  2138    1
## y_rep[522]   122.72    0.13 5.94   115.06   122.80   130.41  2074    1
## y_rep[523]    86.16    0.13 5.85    78.55    86.22    93.52  1974    1
## y_rep[524]   156.05    0.13 5.73   148.80   156.03   163.43  1910    1
## y_rep[525]   156.10    0.13 5.78   148.95   155.85   163.80  1959    1
## y_rep[526]   126.71    0.13 5.76   119.57   126.60   134.41  2023    1
## y_rep[527]   121.10    0.13 5.85   113.78   121.20   128.52  1905    1
## y_rep[528]    98.14    0.12 5.66    90.69    98.25   105.46  2192    1
## y_rep[529]   158.62    0.14 5.69   151.28   158.54   165.82  1644    1
## y_rep[530]   151.94    0.13 5.64   144.65   151.90   159.13  1962    1
## y_rep[531]   126.23    0.12 5.72   118.89   126.23   133.47  2159    1
## y_rep[532]    85.78    0.15 5.81    78.49    85.82    92.97  1470    1
## y_rep[533]   153.42    0.12 5.70   146.12   153.38   160.78  2130    1
## y_rep[534]   153.09    0.13 5.76   145.72   153.16   160.24  2040    1
## y_rep[535]   158.27    0.14 5.63   150.95   158.47   165.33  1726    1
## y_rep[536]   107.75    0.13 5.74   100.34   107.70   115.04  1927    1
## y_rep[537]    74.20    0.13 5.81    66.85    74.07    81.90  1954    1
## y_rep[538]   144.48    0.13 5.76   137.32   144.36   152.03  1911    1
## y_rep[539]    77.16    0.13 5.91    69.68    77.07    84.70  2088    1
## y_rep[540]   139.17    0.13 5.72   131.85   139.21   146.33  1875    1
## y_rep[541]   160.10    0.13 5.80   152.85   160.12   167.48  1859    1
## y_rep[542]   160.40    0.13 5.74   153.02   160.38   167.62  1987    1
## y_rep[543]    77.09    0.14 5.68    69.83    77.07    84.42  1659    1
## y_rep[544]   160.17    0.14 5.74   152.91   160.16   167.41  1715    1
## lp__       -1225.61    0.05 1.44 -1227.50 -1225.27 -1224.12   824    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 15:25:53 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
summary(lm(height~weight.z+I(weight.z^2),data=howell1))
## 
## Call:
## lm(formula = height ~ weight.z + I(weight.z^2), data = howell1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.9689  -3.9794   0.2364   3.9262  19.5182 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   146.6604     0.3748  391.30   <2e-16 ***
## weight.z       21.4149     0.2908   73.64   <2e-16 ***
## I(weight.z^2)  -8.4123     0.2822  -29.80   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.766 on 541 degrees of freedom
## Multiple R-squared:  0.9565, Adjusted R-squared:  0.9564 
## F-statistic:  5952 on 2 and 541 DF,  p-value: < 2.2e-16

Figure 4.9 of Statistical rethinking

6 One continious variable + one categorical variable

6.1 Howell 2: subset with only individuals over 18 years old

ggplot(howell2, aes(y = height, x = weight, group = male)) + geom_point() + geom_smooth(method = "lm")

Xmat <- model.matrix(~weight.z + male, howell2)
m8.data <- with(howell2, list(y = height, X = Xmat, nX = ncol(Xmat), n = nrow(howell2)))
data {
  int<lower=1> n;
  int<lower=1> nX;
  vector [n] y;
  matrix [n,nX] X;
  }
parameters {
  vector[nX] beta;
  real sigma;
  }
  transformed parameters {
  vector[n] mu;

  mu = X*beta;
  }
model {
  y~normal(mu,sigma);
  
  beta ~ normal(0,100);
  sigma~cauchy(0,5);
  }
generated quantities {
  vector[n] log_lik;
  
  for (i in 1:n) {
  log_lik[i] = normal_lpdf(y[i] | mu[i], sigma); 
  }
  }
fit.m8 <- sampling(m8, data = m8.data, iter = 1000, chains = 4, cores = 4)
print(fit.m8, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: ff983e7d557e1f552144a41483b33c4a.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##                 mean se_mean   sd     10%     50%     90% n_eff Rhat
## beta[1]       145.08    0.03 0.81  144.04  145.08  146.14   722    1
## beta[2]         4.14    0.01 0.27    3.80    4.14    4.48   909    1
## beta[3]         6.48    0.02 0.53    5.80    6.49    7.14   725    1
## sigma           4.29    0.00 0.16    4.08    4.28    4.51  1747    1
## mu[1]         159.86    0.01 0.34  159.42  159.86  160.30  1552    1
## mu[2]         146.11    0.01 0.39  145.61  146.11  146.60  1794    1
## mu[3]         143.15    0.01 0.53  142.47  143.15  143.81  1543    1
## mu[4]         163.21    0.01 0.39  162.70  163.20  163.71  1806    1
## mu[5]         149.18    0.01 0.32  148.78  149.18  149.58  1360    1
## mu[6]         169.59    0.02 0.69  168.69  169.60  170.48  1251    1
## mu[7]         147.24    0.01 0.36  146.79  147.24  147.68  1744    1
## mu[8]         164.77    0.01 0.45  164.20  164.77  165.34  1622    1
## mu[9]         145.07    0.01 0.43  144.52  145.07  145.61  1735    1
## mu[10]        164.13    0.01 0.42  163.59  164.14  164.68  1699    1
## mu[11]        154.71    0.02 0.46  154.12  154.70  155.32   760    1
## mu[12]        155.63    0.01 0.45  155.06  155.63  156.21   961    1
## mu[13]        145.82    0.01 0.40  145.31  145.82  146.32  1787    1
## mu[14]        153.30    0.01 0.40  152.80  153.29  153.82   795    1
## mu[15]        144.42    0.01 0.46  143.83  144.42  145.00  1666    1
## mu[16]        160.33    0.01 0.34  159.90  160.34  160.76  1639    1
## mu[17]        156.33    0.01 0.42  155.80  156.33  156.88  1016    1
## mu[18]        147.31    0.01 0.35  146.86  147.31  147.75  1736    1
## mu[19]        160.61    0.01 0.34  160.17  160.61  161.04  1687    1
## mu[20]        150.11    0.01 0.32  149.70  150.11  150.52  1142    1
## mu[21]        151.95    0.02 0.63  151.16  151.97  152.76   852    1
## mu[22]        147.02    0.01 0.36  146.56  147.01  147.47  1765    1
## mu[23]        145.46    0.01 0.42  144.92  145.45  145.98  1770    1
## mu[24]        155.04    0.02 0.48  154.45  155.05  155.66   930    1
## mu[25]        164.53    0.01 0.44  163.97  164.54  165.10  1650    1
## mu[26]        146.76    0.01 0.37  146.29  146.76  147.23  1782    1
## mu[27]        147.40    0.01 0.35  146.96  147.40  147.84  1725    1
## mu[28]        152.60    0.01 0.37  152.13  152.59  153.08   830    1
## mu[29]        147.62    0.01 0.35  147.19  147.62  148.05  1694    1
## mu[30]        145.53    0.01 0.42  145.00  145.53  146.04  1774    1
## mu[31]        153.11    0.01 0.39  152.62  153.10  153.62   803    1
## mu[32]        159.90    0.01 0.34  159.46  159.90  160.34  1559    1
## mu[33]        160.88    0.01 0.34  160.44  160.88  161.31  1730    1
## mu[34]        160.86    0.01 0.34  160.42  160.87  161.30  1727    1
## mu[35]        165.46    0.01 0.48  164.86  165.47  166.07  1543    1
## mu[36]        147.80    0.01 0.34  147.37  147.80  148.23  1665    1
## mu[37]        154.71    0.02 0.46  154.12  154.70  155.32   760    1
## mu[38]        155.64    0.01 0.45  155.08  155.65  156.23   962    1
## mu[39]        147.44    0.01 0.35  147.00  147.44  147.87  1720    1
## mu[40]        156.38    0.01 0.42  155.85  156.38  156.92  1021    1
## mu[41]        157.83    0.01 0.37  157.34  157.83  158.31  1210    1
## mu[42]        166.77    0.01 0.54  166.08  166.78  167.46  1420    1
## mu[43]        152.67    0.01 0.38  152.20  152.67  153.16   826    1
## mu[44]        151.07    0.01 0.33  150.65  151.08  151.50   973    1
## mu[45]        155.35    0.02 0.49  154.72  155.34  156.00   754    1
## mu[46]        157.55    0.02 0.60  156.76  157.53  158.32   748    1
## mu[47]        158.63    0.01 0.35  158.17  158.63  159.08  1334    1
## mu[48]        153.51    0.01 0.41  153.00  153.50  154.04   787    1
## mu[49]        162.68    0.01 0.37  162.19  162.68  163.16  1850    1
## mu[50]        153.25    0.01 0.40  152.76  153.25  153.77   797    1
## mu[51]        151.98    0.01 0.35  151.53  151.98  152.43   875    1
## mu[52]        149.91    0.01 0.32  149.50  149.91  150.32  1185    1
## mu[53]        152.59    0.02 0.60  151.83  152.60  153.36   860    1
## mu[54]        165.06    0.01 0.46  164.48  165.06  165.65  1588    1
## mu[55]        146.56    0.01 0.38  146.08  146.56  147.03  1791    1
## mu[56]        160.17    0.01 0.34  159.73  160.17  160.61  1609    1
## mu[57]        146.66    0.01 0.37  146.18  146.65  147.12  1787    1
## mu[58]        159.44    0.01 0.34  159.00  159.44  159.88  1474    1
## mu[59]        153.04    0.01 0.39  152.55  153.03  153.55   807    1
## mu[60]        166.10    0.01 0.51  165.45  166.10  166.74  1479    1
## mu[61]        147.04    0.01 0.36  146.58  147.03  147.49  1764    1
## mu[62]        152.98    0.01 0.39  152.50  152.98  153.49   809    1
## mu[63]        156.90    0.01 0.40  156.39  156.90  157.42  1077    1
## mu[64]        161.42    0.01 0.35  160.97  161.43  161.88  1800    1
## mu[65]        147.96    0.01 0.34  147.54  147.97  148.39  1635    1
## mu[66]        161.70    0.01 0.35  161.23  161.70  162.15  1824    1
## mu[67]        148.00    0.01 0.34  147.58  148.01  148.43  1628    1
## mu[68]        145.98    0.01 0.40  145.48  145.98  146.48  1792    1
## mu[69]        158.86    0.01 0.35  158.40  158.86  159.31  1372    1
## mu[70]        149.82    0.01 0.32  149.41  149.82  150.23  1206    1
## mu[71]        159.73    0.01 0.34  159.29  159.73  160.17  1528    1
## mu[72]        164.92    0.01 0.45  164.34  164.92  165.50  1604    1
## mu[73]        160.33    0.01 0.34  159.90  160.34  160.76  1639    1
## mu[74]        150.16    0.01 0.32  149.76  150.17  150.58  1131    1
## mu[75]        145.20    0.01 0.43  144.65  145.20  145.73  1747    1
## mu[76]        159.11    0.03 0.69  158.22  159.10  159.99   753    1
## mu[77]        155.58    0.02 0.50  154.94  155.57  156.23   752    1
## mu[78]        159.48    0.01 0.34  159.03  159.48  159.91  1481    1
## mu[79]        148.96    0.01 0.32  148.56  148.97  149.36  1414    1
## mu[80]        143.69    0.01 0.50  143.06  143.69  144.32  1590    1
## mu[81]        165.79    0.01 0.49  165.16  165.80  166.42  1509    1
## mu[82]        156.75    0.01 0.41  156.24  156.75  157.28  1060    1
## mu[83]        148.33    0.01 0.33  147.91  148.33  148.75  1561    1
## mu[84]        158.66    0.01 0.35  158.20  158.66  159.11  1340    1
## mu[85]        149.07    0.01 0.32  148.67  149.08  149.48  1388    1
## mu[86]        159.72    0.01 0.34  159.28  159.72  160.15  1525    1
## mu[87]        146.76    0.01 0.37  146.29  146.76  147.23  1782    1
## mu[88]        158.06    0.01 0.37  157.59  158.06  158.53  1249    1
## mu[89]        149.82    0.01 0.32  149.41  149.82  150.23  1206    1
## mu[90]        164.37    0.01 0.43  163.82  164.37  164.93  1670    1
## mu[91]        163.52    0.01 0.40  163.00  163.51  164.03  1772    1
## mu[92]        162.66    0.01 0.37  162.17  162.66  163.14  1850    1
## mu[93]        149.91    0.01 0.32  149.50  149.91  150.32  1185    1
## mu[94]        154.51    0.02 0.45  153.93  154.50  155.11   763    1
## mu[95]        155.95    0.01 0.44  155.40  155.95  156.52   983    1
## mu[96]        165.12    0.01 0.46  164.53  165.12  165.71  1581    1
## mu[97]        149.75    0.01 0.32  149.34  149.75  150.15  1223    1
## mu[98]        163.24    0.01 0.39  162.73  163.24  163.74  1802    1
## mu[99]        161.41    0.01 0.35  160.95  161.41  161.86  1798    1
## mu[100]       149.98    0.01 0.32  149.58  149.98  150.39  1169    1
## mu[101]       160.68    0.01 0.34  160.24  160.68  161.11  1699    1
## mu[102]       147.40    0.01 0.35  146.96  147.40  147.84  1725    1
## mu[103]       154.65    0.02 0.46  154.06  154.64  155.26   761    1
## mu[104]       167.52    0.02 0.58  166.78  167.52  168.25  1364    1
## mu[105]       159.93    0.01 0.34  159.50  159.94  160.37  1565    1
## mu[106]       147.91    0.01 0.34  147.48  147.92  148.34  1645    1
## mu[107]       146.36    0.01 0.38  145.88  146.36  146.85  1795    1
## mu[108]       149.73    0.01 0.32  149.32  149.73  150.13  1227    1
## mu[109]       157.77    0.01 0.37  157.29  157.77  158.25  1201    1
## mu[110]       153.40    0.01 0.40  152.89  153.39  153.92   791    1
## mu[111]       164.59    0.01 0.44  164.02  164.59  165.15  1643    1
## mu[112]       143.87    0.01 0.49  143.25  143.87  144.49  1608    1
## mu[113]       148.93    0.01 0.32  148.52  148.93  149.33  1423    1
## mu[114]       162.03    0.01 0.36  161.55  162.02  162.48  1843    1
## mu[115]       160.64    0.01 0.34  160.21  160.65  161.07  1693    1
## mu[116]       156.98    0.02 0.57  156.23  156.96  157.72   748    1
## mu[117]       163.88    0.01 0.41  163.35  163.88  164.41  1730    1
## mu[118]       164.70    0.01 0.45  164.13  164.70  165.26  1630    1
## mu[119]       149.80    0.01 0.32  149.39  149.80  150.21  1210    1
## mu[120]       148.96    0.01 0.32  148.56  148.97  149.36  1414    1
## mu[121]       161.06    0.01 0.34  160.62  161.06  161.50  1756    1
## mu[122]       151.15    0.01 0.33  150.72  151.15  151.57   964    1
## mu[123]       158.66    0.01 0.35  158.20  158.66  159.11  1340    1
## mu[124]       149.62    0.01 0.32  149.21  149.62  150.03  1252    1
## mu[125]       155.73    0.02 0.51  155.07  155.72  156.39   751    1
## mu[126]       151.00    0.01 0.33  150.58  151.01  151.43   983    1
## mu[127]       143.36    0.01 0.52  142.70  143.36  144.02  1564    1
## mu[128]       165.59    0.01 0.48  164.97  165.60  166.21  1530    1
## mu[129]       156.49    0.02 0.55  155.79  156.47  157.20   748    1
## mu[130]       146.11    0.01 0.39  145.61  146.11  146.60  1794    1
## mu[131]       160.52    0.01 0.34  160.08  160.52  160.95  1671    1
## mu[132]       165.72    0.01 0.49  165.09  165.72  166.34  1517    1
## mu[133]       149.71    0.01 0.32  149.30  149.71  150.11  1231    1
## mu[134]       161.37    0.01 0.35  160.92  161.37  161.82  1794    1
## mu[135]       152.56    0.01 0.37  152.10  152.56  153.04   832    1
## mu[136]       168.82    0.02 0.65  167.99  168.83  169.65  1287    1
## mu[137]       160.61    0.01 0.34  160.17  160.61  161.04  1687    1
## mu[138]       149.41    0.03 0.78  148.45  149.42  150.39   835    1
## mu[139]       145.35    0.01 0.42  144.81  145.34  145.87  1760    1
## mu[140]       158.46    0.01 0.36  158.00  158.46  158.92  1309    1
## mu[141]       150.85    0.01 0.33  150.44  150.86  151.28  1005    1
## mu[142]       151.93    0.01 0.35  151.48  151.93  152.38   880    1
## mu[143]       147.73    0.01 0.34  147.30  147.73  148.16  1677    1
## mu[144]       149.11    0.01 0.32  148.70  149.11  149.51  1378    1
## mu[145]       151.73    0.01 0.35  151.29  151.73  152.17   898    1
## mu[146]       163.59    0.01 0.40  163.07  163.59  164.11  1764    1
## mu[147]       156.25    0.02 0.53  155.57  156.24  156.95   749    1
## mu[148]       147.78    0.01 0.34  147.35  147.78  148.21  1668    1
## mu[149]       158.50    0.01 0.36  158.03  158.50  158.95  1315    1
## mu[150]       148.62    0.01 0.33  148.21  148.62  149.03  1495    1
## mu[151]       162.21    0.01 0.36  161.73  162.21  162.67  1849    1
## mu[152]       147.51    0.01 0.35  147.07  147.51  147.94  1710    1
## mu[153]       154.35    0.02 0.51  153.72  154.36  155.01   902    1
## mu[154]       151.15    0.01 0.33  150.72  151.15  151.57   964    1
## mu[155]       154.53    0.02 0.50  153.91  154.54  155.18   908    1
## mu[156]       142.64    0.01 0.55  141.94  142.64  143.33  1496    1
## mu[157]       152.71    0.01 0.38  152.24  152.70  153.20   824    1
## mu[158]       155.24    0.02 0.47  154.66  155.25  155.85   939    1
## mu[159]       161.37    0.01 0.35  160.92  161.37  161.82  1794    1
## mu[160]       149.18    0.01 0.32  148.78  149.18  149.58  1360    1
## mu[161]       164.21    0.01 0.43  163.66  164.21  164.75  1690    1
## mu[162]       151.56    0.01 0.34  151.13  151.57  152.00   915    1
## mu[163]       152.15    0.02 0.62  151.37  152.16  152.95   854    1
## mu[164]       158.19    0.01 0.36  157.71  158.19  158.66  1270    1
## mu[165]       160.04    0.01 0.34  159.61  160.04  160.48  1586    1
## mu[166]       152.00    0.01 0.35  151.55  152.00  152.45   873    1
## mu[167]       160.24    0.01 0.34  159.81  160.25  160.68  1623    1
## mu[168]       149.13    0.01 0.32  148.72  149.13  149.53  1374    1
## mu[169]       147.35    0.01 0.35  146.90  147.34  147.79  1732    1
## mu[170]       149.73    0.01 0.32  149.32  149.73  150.13  1227    1
## mu[171]       147.24    0.01 0.36  146.79  147.24  147.68  1744    1
## mu[172]       160.19    0.01 0.34  159.75  160.19  160.62  1613    1
## mu[173]       147.67    0.01 0.34  147.24  147.68  148.10  1686    1
## mu[174]       154.86    0.02 0.48  154.26  154.87  155.50   921    1
## mu[175]       154.91    0.02 0.47  154.31  154.90  155.53   758    1
## mu[176]       164.01    0.01 0.42  163.47  164.01  164.55  1714    1
## mu[177]       151.95    0.01 0.35  151.51  151.96  152.41   877    1
## mu[178]       148.00    0.01 0.34  147.58  148.01  148.43  1628    1
## mu[179]       156.90    0.01 0.40  156.39  156.90  157.42  1077    1
## mu[180]       143.15    0.01 0.53  142.47  143.15  143.81  1543    1
## mu[181]       158.33    0.01 0.36  157.87  158.33  158.80  1291    1
## mu[182]       147.11    0.01 0.36  146.65  147.11  147.56  1757    1
## mu[183]       145.86    0.01 0.40  145.35  145.85  146.35  1788    1
## mu[184]       148.93    0.01 0.32  148.52  148.93  149.33  1423    1
## mu[185]       159.92    0.01 0.34  159.48  159.92  160.36  1562    1
## mu[186]       159.79    0.01 0.34  159.35  159.79  160.23  1538    1
## mu[187]       152.45    0.01 0.37  151.99  152.45  152.93   839    1
## mu[188]       155.59    0.01 0.45  155.02  155.59  156.18   959    1
## mu[189]       146.18    0.01 0.39  145.69  146.18  146.67  1795    1
## mu[190]       160.06    0.01 0.34  159.62  160.06  160.50  1589    1
## mu[191]       146.26    0.01 0.39  145.76  146.25  146.74  1795    1
## mu[192]       165.46    0.01 0.48  164.86  165.47  166.07  1543    1
## mu[193]       147.53    0.01 0.35  147.09  147.53  147.96  1708    1
## mu[194]       159.64    0.01 0.34  159.20  159.64  160.08  1511    1
## mu[195]       145.93    0.01 0.40  145.42  145.92  146.42  1790    1
## mu[196]       149.18    0.01 0.32  148.78  149.18  149.58  1360    1
## mu[197]       151.42    0.01 0.34  150.99  151.42  151.85   931    1
## mu[198]       161.53    0.01 0.35  161.08  161.54  161.99  1811    1
## mu[199]       164.64    0.01 0.44  164.08  164.64  165.21  1637    1
## mu[200]       147.04    0.01 0.36  146.58  147.03  147.49  1764    1
## mu[201]       166.68    0.01 0.54  166.00  166.69  167.36  1427    1
## mu[202]       151.20    0.01 0.34  150.78  151.21  151.63   957    1
## mu[203]       157.66    0.01 0.38  157.18  157.67  158.15  1184    1
## mu[204]       151.29    0.01 0.34  150.86  151.30  151.72   945    1
## mu[205]       157.81    0.01 0.37  157.33  157.81  158.29  1207    1
## mu[206]       148.67    0.01 0.33  148.27  148.67  149.08  1483    1
## mu[207]       150.98    0.01 0.33  150.56  150.99  151.41   986    1
## mu[208]       144.78    0.01 0.45  144.21  144.78  145.34  1705    1
## mu[209]       143.73    0.01 0.50  143.10  143.73  144.35  1593    1
## mu[210]       145.76    0.01 0.41  145.25  145.76  146.27  1785    1
## mu[211]       160.97    0.01 0.34  160.53  160.97  161.41  1744    1
## mu[212]       163.30    0.01 0.39  162.79  163.30  163.80  1796    1
## mu[213]       152.97    0.02 0.58  152.24  152.98  153.71   867    1
## mu[214]       148.69    0.01 0.33  148.29  148.69  149.10  1479    1
## mu[215]       161.64    0.01 0.35  161.18  161.64  162.10  1820    1
## mu[216]       157.39    0.01 0.38  156.89  157.39  157.89  1144    1
## mu[217]       161.19    0.01 0.34  160.74  161.19  161.63  1773    1
## mu[218]       144.26    0.01 0.47  143.65  144.25  144.85  1649    1
## mu[219]       157.32    0.01 0.39  156.82  157.32  157.82  1133    1
## mu[220]       153.58    0.01 0.41  153.07  153.57  154.12   785    1
## mu[221]       150.13    0.01 0.32  149.72  150.13  150.54  1138    1
## mu[222]       144.46    0.01 0.46  143.86  144.45  145.04  1670    1
## mu[223]       154.66    0.02 0.49  154.05  154.67  155.30   913    1
## mu[224]       145.76    0.01 0.41  145.25  145.76  146.27  1785    1
## mu[225]       161.84    0.01 0.35  161.38  161.84  162.30  1834    1
## mu[226]       152.05    0.01 0.36  151.60  152.05  152.51   869    1
## mu[227]       154.27    0.02 0.44  153.71  154.26  154.85   767    1
## mu[228]       166.92    0.01 0.55  166.22  166.93  167.61  1408    1
## mu[229]       150.58    0.01 0.33  150.16  150.58  151.00  1051    1
## mu[230]       162.52    0.01 0.37  162.03  162.51  162.99  1852    1
## mu[231]       161.70    0.01 0.35  161.23  161.70  162.15  1824    1
## mu[232]       144.67    0.01 0.45  144.09  144.67  145.24  1694    1
## mu[233]       147.96    0.01 0.34  147.54  147.97  148.39  1635    1
## mu[234]       167.39    0.02 0.57  166.66  167.39  168.12  1373    1
## mu[235]       162.64    0.01 0.37  162.15  162.64  163.12  1850    1
## mu[236]       148.35    0.01 0.33  147.93  148.35  148.77  1557    1
## mu[237]       157.37    0.01 0.39  156.88  157.37  157.87  1141    1
## mu[238]       157.73    0.02 0.61  156.93  157.71  158.51   748    1
## mu[239]       151.62    0.01 0.34  151.18  151.62  152.06   909    1
## mu[240]       149.22    0.01 0.32  148.82  149.22  149.62  1351    1
## mu[241]       149.58    0.01 0.32  149.18  149.58  149.99  1261    1
## mu[242]       156.46    0.01 0.42  155.93  156.46  157.00  1029    1
## mu[243]       156.86    0.01 0.40  156.35  156.86  157.38  1073    1
## mu[244]       148.89    0.01 0.32  148.49  148.89  149.29  1431    1
## mu[245]       156.68    0.01 0.41  156.16  156.68  157.21  1052    1
## mu[246]       152.35    0.01 0.36  151.89  152.34  152.81   847    1
## mu[247]       159.88    0.01 0.34  159.44  159.88  160.32  1555    1
## mu[248]       149.98    0.01 0.32  149.58  149.98  150.39  1169    1
## mu[249]       160.30    0.01 0.34  159.86  160.30  160.73  1633    1
## mu[250]       158.61    0.01 0.35  158.15  158.61  159.06  1331    1
## mu[251]       163.12    0.01 0.39  162.61  163.11  163.61  1815    1
## mu[252]       159.70    0.01 0.34  159.26  159.70  160.13  1521    1
## mu[253]       150.64    0.01 0.33  150.22  150.64  151.06  1041    1
## mu[254]       150.56    0.01 0.33  150.14  150.57  150.98  1054    1
## mu[255]       156.84    0.02 0.56  156.10  156.82  157.56   748    1
## mu[256]       151.36    0.01 0.34  150.93  151.37  151.80   937    1
## mu[257]       159.46    0.01 0.34  159.01  159.46  159.90  1477    1
## mu[258]       143.02    0.01 0.53  142.34  143.02  143.69  1531    1
## mu[259]       165.39    0.01 0.48  164.79  165.39  166.00  1551    1
## mu[260]       164.97    0.01 0.46  164.39  164.97  165.56  1598    1
## mu[261]       159.64    0.01 0.34  159.20  159.64  160.08  1511    1
## mu[262]       151.67    0.01 0.35  151.23  151.68  152.12   903    1
## mu[263]       151.91    0.01 0.35  151.46  151.91  152.36   881    1
## mu[264]       154.05    0.02 0.43  153.51  154.04  154.62   771    1
## mu[265]       159.06    0.01 0.35  158.61  159.06  159.50  1406    1
## mu[266]       150.85    0.01 0.33  150.44  150.86  151.28  1005    1
## mu[267]       165.13    0.01 0.46  164.55  165.14  165.73  1579    1
## mu[268]       148.42    0.01 0.33  148.01  148.42  148.83  1541    1
## mu[269]       155.29    0.02 0.49  154.66  155.28  155.93   754    1
## mu[270]       158.88    0.01 0.35  158.43  158.88  159.33  1375    1
## mu[271]       160.84    0.01 0.34  160.41  160.85  161.28  1725    1
## mu[272]       151.00    0.01 0.33  150.58  151.01  151.43   983    1
## mu[273]       161.92    0.01 0.36  161.45  161.91  162.37  1838    1
## mu[274]       155.33    0.02 0.46  154.76  155.34  155.94   944    1
## mu[275]       146.80    0.01 0.37  146.33  146.80  147.26  1780    1
## mu[276]       162.28    0.01 0.36  161.80  162.28  162.74  1851    1
## mu[277]       151.45    0.01 0.34  151.02  151.46  151.89   926    1
## mu[278]       150.55    0.01 0.33  150.13  150.55  150.96  1057    1
## mu[279]       159.15    0.01 0.35  158.70  159.16  159.59  1422    1
## mu[280]       148.07    0.01 0.34  147.66  148.08  148.50  1614    1
## mu[281]       145.02    0.01 0.44  144.46  145.02  145.56  1729    1
## mu[282]       154.39    0.02 0.51  153.76  154.40  155.04   903    1
## mu[283]       146.73    0.01 0.37  146.25  146.73  147.19  1784    1
## mu[284]       160.41    0.01 0.34  159.97  160.41  160.83  1652    1
## mu[285]       145.55    0.01 0.41  145.02  145.54  146.06  1775    1
## mu[286]       157.23    0.01 0.39  156.72  157.23  157.73  1119    1
## mu[287]       160.10    0.01 0.34  159.66  160.10  160.53  1596    1
## mu[288]       161.88    0.01 0.36  161.41  161.88  162.33  1836    1
## mu[289]       150.93    0.01 0.33  150.51  150.93  151.35   994    1
## mu[290]       160.10    0.01 0.34  159.66  160.10  160.53  1596    1
## mu[291]       159.13    0.01 0.35  158.68  159.14  159.57  1418    1
## mu[292]       165.37    0.01 0.47  164.77  165.37  165.98  1553    1
## mu[293]       160.35    0.01 0.34  159.91  160.35  160.78  1642    1
## mu[294]       153.64    0.01 0.41  153.12  153.63  154.17   783    1
## mu[295]       159.64    0.01 0.34  159.20  159.64  160.08  1511    1
## mu[296]       145.51    0.01 0.42  144.98  145.51  146.03  1773    1
## mu[297]       146.18    0.01 0.39  145.69  146.18  146.67  1795    1
## mu[298]       154.11    0.02 0.43  153.56  154.10  154.68   770    1
## mu[299]       162.06    0.01 0.36  161.59  162.06  162.52  1845    1
## mu[300]       159.02    0.01 0.35  158.57  159.03  159.46  1399    1
## mu[301]       146.42    0.01 0.38  145.93  146.42  146.90  1794    1
## mu[302]       160.39    0.01 0.34  159.95  160.39  160.82  1649    1
## mu[303]       147.11    0.01 0.36  146.65  147.11  147.56  1757    1
## mu[304]       159.52    0.01 0.34  159.07  159.52  159.95  1487    1
## mu[305]       164.72    0.01 0.45  164.14  164.72  165.28  1628    1
## mu[306]       163.95    0.01 0.42  163.42  163.95  164.49  1721    1
## mu[307]       160.28    0.01 0.34  159.84  160.28  160.71  1629    1
## mu[308]       150.33    0.01 0.32  149.91  150.33  150.74  1098    1
## mu[309]       144.64    0.01 0.45  144.05  144.64  145.21  1690    1
## mu[310]       154.75    0.02 0.46  154.15  154.73  155.36   760    1
## mu[311]       151.15    0.01 0.33  150.72  151.15  151.57   964    1
## mu[312]       144.16    0.01 0.48  143.56  144.16  144.76  1639    1
## mu[313]       159.52    0.01 0.34  159.07  159.52  159.95  1487    1
## mu[314]       152.29    0.01 0.36  151.83  152.29  152.76   851    1
## mu[315]       159.59    0.01 0.34  159.14  159.59  160.02  1501    1
## mu[316]       148.91    0.01 0.32  148.51  148.91  149.31  1427    1
## mu[317]       161.48    0.01 0.35  161.02  161.48  161.93  1805    1
## mu[318]       151.65    0.01 0.35  151.22  151.66  152.10   905    1
## mu[319]       156.28    0.01 0.42  155.74  156.28  156.83  1011    1
## mu[320]       149.42    0.01 0.32  149.01  149.42  149.82  1301    1
## mu[321]       158.19    0.01 0.36  157.71  158.19  158.66  1270    1
## mu[322]       155.58    0.02 0.50  154.94  155.57  156.23   752    1
## mu[323]       160.79    0.01 0.34  160.35  160.79  161.22  1716    1
## mu[324]       162.02    0.01 0.36  161.55  162.02  162.48  1843    1
## mu[325]       150.84    0.01 0.33  150.42  150.84  151.26  1008    1
## mu[326]       145.49    0.01 0.42  144.96  145.49  146.01  1772    1
## mu[327]       150.21    0.01 0.32  149.80  150.21  150.63  1121    1
## mu[328]       159.19    0.01 0.35  158.74  159.19  159.62  1428    1
## mu[329]       156.04    0.01 0.43  155.50  156.04  156.61   990    1
## mu[330]       153.69    0.02 0.54  153.01  153.70  154.39   882    1
## mu[331]       156.96    0.01 0.40  156.45  156.96  157.47  1084    1
## mu[332]       163.59    0.01 0.40  163.07  163.59  164.11  1764    1
## mu[333]       150.27    0.01 0.32  149.86  150.27  150.69  1109    1
## mu[334]       154.70    0.02 0.49  154.09  154.70  155.34   915    1
## mu[335]       150.51    0.01 0.33  150.09  150.51  150.93  1064    1
## mu[336]       162.97    0.01 0.38  162.47  162.97  163.46  1828    1
## mu[337]       147.36    0.01 0.35  146.92  147.36  147.81  1730    1
## mu[338]       150.16    0.01 0.32  149.76  150.17  150.58  1131    1
## mu[339]       145.73    0.01 0.41  145.21  145.73  146.23  1783    1
## mu[340]       151.07    0.01 0.33  150.65  151.08  151.50   973    1
## mu[341]       147.13    0.01 0.36  146.67  147.12  147.58  1756    1
## mu[342]       157.48    0.01 0.38  156.99  157.48  157.98  1158    1
## mu[343]       150.95    0.01 0.33  150.52  150.95  151.37   991    1
## mu[344]       159.90    0.01 0.34  159.46  159.90  160.34  1559    1
## mu[345]       147.98    0.01 0.34  147.56  147.99  148.41  1632    1
## mu[346]       155.52    0.01 0.46  154.95  155.52  156.11   954    1
## mu[347]       148.89    0.01 0.32  148.49  148.89  149.29  1431    1
## mu[348]       152.87    0.01 0.38  152.40  152.87  153.37   815    1
## mu[349]       144.67    0.01 0.45  144.09  144.67  145.24  1694    1
## mu[350]       162.64    0.01 0.37  162.15  162.64  163.12  1850    1
## mu[351]       157.38    0.02 0.59  156.60  157.37  158.14   748    1
## mu[352]       162.88    0.01 0.38  162.38  162.88  163.37  1836    1
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## log_lik[30]    -2.57    0.00 0.06   -2.66   -2.57   -2.49  1859    1
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## log_lik[37]    -3.63    0.01 0.18   -3.86   -3.62   -3.40   813    1
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## log_lik[295]   -2.40    0.00 0.04   -2.46   -2.40   -2.35  1711    1
## log_lik[296]   -2.47    0.00 0.05   -2.54   -2.47   -2.40  1618    1
## log_lik[297]   -2.39    0.00 0.04   -2.44   -2.38   -2.34  1592    1
## log_lik[298]   -3.64    0.01 0.17   -3.86   -3.63   -3.42   832    1
## log_lik[299]   -5.65    0.01 0.30   -6.04   -5.64   -5.26  1682    1
## log_lik[300]   -2.40    0.00 0.04   -2.45   -2.40   -2.35  1610    1
## log_lik[301]   -2.40    0.00 0.04   -2.45   -2.40   -2.35  1586    1
## log_lik[302]   -3.23    0.00 0.11   -3.38   -3.23   -3.10  1603    1
## log_lik[303]   -2.58    0.00 0.06   -2.66   -2.58   -2.51  1682    1
## log_lik[304]   -2.54    0.00 0.05   -2.61   -2.53   -2.47  1545    1
## log_lik[305]   -2.40    0.00 0.04   -2.46   -2.40   -2.35  1531    1
## log_lik[306]   -2.80    0.00 0.09   -2.92   -2.80   -2.69  1640    1
## log_lik[307]   -3.16    0.00 0.10   -3.29   -3.15   -3.03  1600    1
## log_lik[308]   -2.49    0.00 0.05   -2.56   -2.49   -2.43  1311    1
## log_lik[309]   -2.43    0.00 0.05   -2.50   -2.43   -2.37  1559    1
## log_lik[310]   -2.59    0.00 0.07   -2.68   -2.58   -2.50   784    1
## log_lik[311]   -2.39    0.00 0.04   -2.44   -2.39   -2.34  1570    1
## log_lik[312]   -2.39    0.00 0.04   -2.44   -2.39   -2.34  1519    1
## log_lik[313]   -2.41    0.00 0.04   -2.47   -2.41   -2.36  1689    1
## log_lik[314]   -2.41    0.00 0.04   -2.47   -2.41   -2.36  1252    1
## log_lik[315]   -3.33    0.00 0.12   -3.48   -3.33   -3.19  1493    1
## log_lik[316]   -2.45    0.00 0.04   -2.50   -2.44   -2.39  1588    1
## log_lik[317]  -15.34    0.02 1.02  -16.67  -15.31  -14.04  1814    1
## log_lik[318]   -3.30    0.00 0.11   -3.45   -3.30   -3.16   991    1
## log_lik[319]   -2.43    0.00 0.05   -2.49   -2.42   -2.37  1368    1
## log_lik[320]   -3.33    0.00 0.11   -3.47   -3.33   -3.19  1329    1
## log_lik[321]   -2.38    0.00 0.04   -2.43   -2.38   -2.33  1711    1
## log_lik[322]   -4.25    0.01 0.25   -4.56   -4.23   -3.93   847    1
## log_lik[323]   -2.39    0.00 0.04   -2.45   -2.39   -2.34  1651    1
## log_lik[324]   -2.64    0.00 0.06   -2.72   -2.63   -2.56  1872    1
## log_lik[325]   -2.84    0.00 0.07   -2.93   -2.83   -2.74  1056    1
## log_lik[326]   -4.21    0.00 0.21   -4.47   -4.20   -3.94  1963    1
## log_lik[327]   -4.40    0.01 0.19   -4.65   -4.40   -4.15  1320    1
## log_lik[328]   -2.97    0.00 0.09   -3.08   -2.96   -2.86  1409    1
## log_lik[329]   -3.13    0.00 0.13   -3.29   -3.12   -2.97   989    1
## log_lik[330]   -2.42    0.00 0.05   -2.49   -2.42   -2.36  1078    1
## log_lik[331]   -2.89    0.00 0.09   -3.02   -2.89   -2.77  1041    1
## log_lik[332]   -3.57    0.00 0.15   -3.77   -3.56   -3.37  1718    1
## log_lik[333]   -2.38    0.00 0.04   -2.43   -2.38   -2.33  1644    1
## log_lik[334]   -3.35    0.01 0.16   -3.56   -3.34   -3.14   939    1
## log_lik[335]   -2.91    0.00 0.08   -3.02   -2.91   -2.81  1121    1
## log_lik[336]   -2.69    0.00 0.07   -2.79   -2.69   -2.60  1852    1
## log_lik[337]   -2.40    0.00 0.04   -2.45   -2.40   -2.35  1594    1
## log_lik[338]   -2.45    0.00 0.04   -2.50   -2.45   -2.39  1406    1
## log_lik[339]   -2.60    0.00 0.07   -2.69   -2.60   -2.52  1690    1
## log_lik[340]   -2.56    0.00 0.05   -2.63   -2.56   -2.49  1121    1
## log_lik[341]   -2.38    0.00 0.04   -2.43   -2.38   -2.33  1715    1
## log_lik[342]   -2.55    0.00 0.06   -2.63   -2.55   -2.48  1241    1
## log_lik[343]   -3.13    0.00 0.10   -3.26   -3.13   -3.01  1064    1
## log_lik[344]   -2.39    0.00 0.04   -2.44   -2.39   -2.34  1739    1
## log_lik[345]   -2.48    0.00 0.05   -2.54   -2.48   -2.42  1699    1
## log_lik[346]   -2.39    0.00 0.04   -2.45   -2.39   -2.34  1476    1
## log_lik[347]   -2.71    0.00 0.06   -2.80   -2.71   -2.63  1464    1
## log_lik[348]   -4.94    0.01 0.25   -5.27   -4.93   -4.62  1051    1
## log_lik[349]   -2.47    0.00 0.05   -2.54   -2.46   -2.40  1847    1
## log_lik[350]   -2.38    0.00 0.04   -2.43   -2.38   -2.33  1683    1
## log_lik[351]   -2.42    0.00 0.05   -2.49   -2.42   -2.36   910    1
## log_lik[352]   -2.84    0.00 0.09   -2.96   -2.84   -2.74  1745    1
## lp__         -689.28    0.05 1.42 -691.14 -688.97 -687.74   750    1
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 15:25:55 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
summary(lm(height~weight.z+male,data=howell2))
## 
## Call:
## lm(formula = height ~ weight.z + male, data = howell2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.7859  -2.5506   0.4669   2.6278  14.1486 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 145.0498     0.8194  177.02   <2e-16 ***
## weight.z      4.1399     0.2678   15.46   <2e-16 ***
## male          6.5003     0.5359   12.13   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.272 on 349 degrees of freedom
## Multiple R-squared:  0.6973, Adjusted R-squared:  0.6955 
## F-statistic: 401.9 on 2 and 349 DF,  p-value: < 2.2e-16

6.2 Howell 1: data set with all individuals

howell1$male <- ifelse(howell1$male==1,2,1)
m8.1.data <- list(N = nrow(howell1), height = howell1$height, weight = howell1$weight.z, male = as.integer(howell1$male), n = length(unique(howell1$male)))
data {                                                     // observed variables 
  int<lower=1> N;                                          // number of observations
  vector[N] height;                                        // response variables
  vector[N] weight;                                        // predictor (previously centered)
  int<lower=0> n; 
  int<lower=1,upper=n> male[N]; 
}

parameters {                                               // unobserved variables
  real beta[2];                                            // parameters: intercept and slope
  vector[n] alpha; 
  real sigma;
}

transformed parameters {
vector[N] mu;                                              // create one variable with predictions for each obs
for (i in 1:N)                                             // loop for all cases 
mu[i] = alpha[male[i]] + beta[1]*weight[i] + beta[2]*(weight[i] .* weight[i]);  
}


model {
  height ~ normal(mu,sigma);        // likelihood
  alpha ~ normal(178, 100);                              // prior of the intercept
  beta[1] ~ normal(0, 100);                                  // prior of the slope
  beta[2] ~ normal(0, 100);
  sigma ~ uniform(0,50);
}
fit.m8.1 <- sampling(m8.1, data = m8.1.data, iter = 1000, chains = 4, cores = 4)
print(fit.m8.1, probs = c(0.10, 0.5, 0.9))
## Inference for Stan model: 7f6f4063a593c9d5515aafcd1d091662.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##              mean se_mean   sd      10%      50%      90% n_eff Rhat
## beta[1]     20.68    0.01 0.29    20.30    20.67    21.05   650 1.00
## beta[2]     -9.13    0.01 0.28    -9.49    -9.13    -8.75   724 1.00
## alpha[1]   145.32    0.01 0.38   144.84   145.33   145.81  1517 1.00
## alpha[2]   149.67    0.02 0.50   149.03   149.66   150.30  1016 1.00
## sigma        5.41    0.00 0.17     5.19     5.40     5.62  2199 1.00
## mu[1]      160.54    0.01 0.36   160.07   160.53   161.02  1845 1.00
## mu[2]      146.52    0.01 0.38   146.04   146.52   146.99  1606 1.00
## mu[3]      139.47    0.01 0.39   138.97   139.46   139.98  1267 1.00
## mu[4]      161.35    0.02 0.47   160.75   161.34   161.97   899 1.00
## mu[5]      151.93    0.01 0.36   151.48   151.93   152.38  2018 1.00
## mu[6]      156.53    0.04 1.07   155.13   156.52   157.93   614 1.00
## mu[7]      148.73    0.01 0.37   148.26   148.74   149.19  1824 1.00
## mu[8]      160.94    0.02 0.57   160.20   160.93   161.70   735 1.00
## mu[9]      144.26    0.01 0.38   143.77   144.27   144.76  1451 1.00
## mu[10]     161.17    0.02 0.53   160.49   161.16   161.87   786 1.00
## mu[11]     156.79    0.01 0.47   156.19   156.77   157.40  1030 1.00
## mu[12]     156.22    0.01 0.42   155.68   156.21   156.76  1434 1.00
## mu[13]     145.91    0.01 0.38   145.42   145.91   146.39  1558 1.00
## mu[14]     156.15    0.01 0.41   155.63   156.14   156.68  1367 1.00
## mu[15]     142.72    0.01 0.39   142.23   142.72   143.22  1375 1.00
## mu[16]     160.79    0.01 0.37   160.33   160.78   161.28  1690 1.00
## mu[17]     157.20    0.01 0.41   156.68   157.19   157.72  1573 1.00
## mu[18]     148.86    0.01 0.37   148.40   148.87   149.32  1840 1.00
## mu[19]     116.42    0.02 0.49   115.78   116.42   117.08   983 1.00
## mu[20]      95.12    0.01 0.47    94.54    95.11    95.71  2330 1.00
## mu[21]      83.45    0.01 0.58    82.71    83.44    84.18  2332 1.00
## mu[22]     160.92    0.01 0.37   160.44   160.90   161.40  1593 1.00
## mu[23]     153.18    0.01 0.36   152.74   153.18   153.63  2025 1.00
## mu[24]     126.69    0.02 0.51   126.01   126.68   127.35   879 1.00
## mu[25]     101.54    0.01 0.42   101.00   101.53   102.07  2081 1.00
## mu[26]     149.50    0.02 0.50   148.87   149.49   150.14  1011 1.00
## mu[27]     148.32    0.01 0.37   147.85   148.33   148.78  1778 1.00
## mu[28]     145.12    0.01 0.38   144.63   145.12   145.61  1503 1.00
## mu[29]     135.14    0.02 0.53   134.46   135.13   135.82   850 1.00
## mu[30]     120.11    0.01 0.38   119.63   120.09   120.60  1285 1.00
## mu[31]     111.46    0.01 0.49   110.82   111.45   112.12  1066 1.00
## mu[32]     155.34    0.01 0.44   154.78   155.33   155.89  1339 1.00
## mu[33]     161.03    0.02 0.56   160.32   161.03   161.77   752 1.00
## mu[34]     147.84    0.01 0.37   147.36   147.84   148.30  1726 1.00
## mu[35]     149.03    0.01 0.37   148.56   149.03   149.49  1859 1.00
## mu[36]     155.68    0.01 0.39   155.19   155.66   156.18  1563 1.00
## mu[37]     149.42    0.01 0.36   148.96   149.42   149.88  1899 1.00
## mu[38]     145.28    0.01 0.38   144.79   145.28   145.76  1513 1.00
## mu[39]     156.03    0.01 0.41   155.52   156.02   156.56  1429 1.00
## mu[40]     160.56    0.01 0.36   160.09   160.55   161.04  1834 1.00
## mu[41]     161.03    0.01 0.38   160.55   161.01   161.52  1495 1.00
## mu[42]     161.02    0.01 0.38   160.54   161.00   161.51  1502 1.00
## mu[43]     160.60    0.02 0.63   159.80   160.59   161.43   696 1.00
## mu[44]     149.74    0.01 0.36   149.28   149.74   150.20  1916 1.00
## mu[45]     156.79    0.01 0.47   156.19   156.77   157.40  1030 1.00
## mu[46]     133.81    0.01 0.39   133.31   133.79   134.31  1179 1.00
## mu[47]     118.58    0.02 0.50   117.94   118.58   119.24   957 1.00
## mu[48]     121.79    0.01 0.38   121.31   121.77   122.27  1248 1.00
## mu[49]      92.90    0.01 0.48    92.29    92.90    93.52  2375 1.00
## mu[50]     156.25    0.01 0.42   155.70   156.24   156.79  1437 1.00
## mu[51]     149.10    0.01 0.37   148.63   149.10   149.55  1866 1.00
## mu[52]     157.26    0.01 0.41   156.74   157.25   157.78  1583 1.00
## mu[53]     158.92    0.01 0.38   158.43   158.91   159.42  1904 1.00
## mu[54]     122.77    0.02 0.51   122.11   122.77   123.43   917 1.00
## mu[55]      99.79    0.01 0.43    99.25    99.78   100.33  2163 1.00
## mu[56]      91.22    0.01 0.50    90.59    91.21    91.84  2393 1.00
## mu[57]     159.69    0.03 0.75   158.73   159.67   160.67   651 1.00
## mu[58]     155.73    0.01 0.39   155.24   155.71   156.24  1543 1.00
## mu[59]     154.30    0.01 0.36   153.84   154.29   154.76  1956 1.00
## mu[60]     156.95    0.02 0.51   156.32   156.94   157.61   932 1.00
## mu[61]     156.86    0.02 0.66   156.03   156.85   157.75   740 1.00
## mu[62]     159.66    0.01 0.37   159.18   159.65   160.14  2008 1.00
## mu[63]     156.27    0.01 0.42   155.74   156.26   156.81  1304 1.00
## mu[64]     161.37    0.01 0.44   160.82   161.36   161.95   992 1.00
## mu[65]     156.12    0.01 0.41   155.60   156.11   156.65  1382 1.00
## mu[66]     155.18    0.01 0.38   154.71   155.16   155.66  1741 1.00
## mu[67]     152.93    0.01 0.36   152.48   152.92   153.37  2030 1.00
## mu[68]     150.86    0.01 0.49   150.25   150.85   151.49  1065 1.00
## mu[69]     160.81    0.02 0.60   160.04   160.80   161.60   717 1.00
## mu[70]     147.44    0.01 0.37   146.97   147.45   147.91  1688 1.00
## mu[71]     128.98    0.02 0.52   128.31   128.97   129.65   864 1.00
## mu[72]     127.10    0.01 0.39   126.62   127.09   127.60  1178 1.00
## mu[73]     160.71    0.01 0.37   160.25   160.69   161.19  1747 1.00
## mu[74]     147.62    0.01 0.37   147.15   147.63   148.09  1705 1.00
## mu[75]     135.35    0.01 0.39   134.86   135.34   135.85  1193 1.00
## mu[76]     160.27    0.01 0.36   159.80   160.27   160.75  1948 1.00
## mu[77]     155.98    0.01 0.40   155.48   155.97   156.51  1447 1.00
## mu[78]      96.12    0.01 0.52    95.43    96.13    96.80  1452 1.00
## mu[79]     160.20    0.03 0.69   159.32   160.19   161.10   671 1.00
## mu[80]     148.36    0.01 0.37   147.89   148.36   148.82  1782 1.00
## mu[81]     152.49    0.01 0.36   152.04   152.49   152.94  2029 1.00
## mu[82]     131.34    0.01 0.39   130.84   131.32   131.83  1167 1.00
## mu[83]     155.94    0.01 0.40   155.44   155.93   156.47  1461 1.00
## mu[84]     157.90    0.01 0.39   157.40   157.90   158.42  1698 1.00
## mu[85]     161.20    0.01 0.39   160.71   161.18   161.71  1313 1.00
## mu[86]     150.02    0.01 0.36   149.56   150.02   150.48  1931 1.00
## mu[87]     110.30    0.01 0.49   109.67   110.29   110.96  1085 1.00
## mu[88]     161.27    0.01 0.40   160.75   161.24   161.79  1232 1.00
## mu[89]     150.08    0.01 0.36   149.62   150.08   150.54  1935 1.00
## mu[90]     146.26    0.01 0.38   145.77   146.26   146.73  1584 1.00
## mu[91]     159.85    0.01 0.37   159.37   159.84   160.33  2011 1.00
## mu[92]     152.81    0.01 0.36   152.36   152.80   153.25  2030 1.00
## mu[93]     160.46    0.01 0.36   159.99   160.45   160.94  1880 1.00
## mu[94]     160.88    0.02 0.59   160.13   160.87   161.65   726 1.00
## mu[95]     109.87    0.01 0.39   109.38   109.86   110.36  1660 1.00
## mu[96]     160.79    0.01 0.37   160.33   160.78   161.28  1690 1.00
## mu[97]     153.25    0.01 0.36   152.80   153.25   153.70  2023 1.00
## mu[98]     144.55    0.01 0.38   144.06   144.56   145.04  1468 1.00
## mu[99]     111.54    0.01 0.49   110.90   111.54   112.20  1064 1.00
## mu[100]    156.19    0.03 0.80   155.16   156.18   157.27   679 1.00
## mu[101]    135.38    0.02 0.53   134.70   135.37   136.05   850 1.00
## mu[102]    156.99    0.02 0.52   156.33   156.97   157.67   902 1.00
## mu[103]    160.30    0.01 0.36   159.83   160.29   160.78  1941 1.00
## mu[104]    151.61    0.01 0.36   151.16   151.61   152.07  2007 1.00
## mu[105]    140.90    0.01 0.39   140.41   140.90   141.41  1308 1.00
## mu[106]    160.41    0.03 0.66   159.57   160.39   161.27   682 1.00
## mu[107]    157.73    0.01 0.40   157.22   157.72   158.24  1665 1.00
## mu[108]    150.62    0.01 0.36   150.16   150.62   151.08  1963 1.00
## mu[109]    140.48    0.01 0.39   139.99   140.47   140.99  1294 1.00
## mu[110]    109.30    0.01 0.49   108.68   109.29   109.95  1104 1.00
## mu[111]    107.43    0.01 0.49   106.80   107.43   108.08  1142 1.00
## mu[112]    159.69    0.01 0.37   159.21   159.68   160.17  2010 1.00
## mu[113]    151.77    0.01 0.36   151.32   151.77   152.23  2013 1.00
## mu[114]    108.26    0.01 0.40   107.77   108.25   108.75  1737 1.00
## mu[115]    103.07    0.01 0.50   102.42   103.07   103.71  1247 1.00
## mu[116]     91.24    0.01 0.55    90.53    91.23    91.94  1600 1.00
## mu[117]    160.45    0.01 0.36   159.98   160.44   160.93  1885 1.00
## mu[118]    147.84    0.01 0.37   147.36   147.84   148.30  1726 1.00
## mu[119]    159.15    0.01 0.37   158.67   159.14   159.65  1947 1.00
## mu[120]    152.81    0.01 0.36   152.36   152.80   153.25  2030 1.00
## mu[121]    161.09    0.02 0.54   160.39   161.09   161.82   765 1.00
## mu[122]    161.31    0.02 0.49   160.69   161.30   161.96   855 1.00
## mu[123]    161.37    0.01 0.44   160.82   161.36   161.95   995 1.00
## mu[124]     85.17    0.01 0.59    84.41    85.17    85.93  1750 1.00
## mu[125]    152.93    0.01 0.36   152.48   152.92   153.37  2030 1.00
## mu[126]    156.72    0.01 0.46   156.14   156.70   157.32  1066 1.00
## mu[127]     96.69    0.01 0.52    96.00    96.69    97.36  1434 1.00
## mu[128]    156.68    0.01 0.42   156.15   156.68   157.22  1495 1.00
## mu[129]    160.78    0.02 0.60   160.01   160.77   161.58   714 1.00
## mu[130]    152.71    0.01 0.36   152.26   152.71   153.16  2031 1.00
## mu[131]    111.46    0.01 0.39   110.97   111.44   111.94  1590 1.00
## mu[132]    103.07    0.01 0.50   102.42   103.07   103.71  1247 1.00
## mu[133]    161.34    0.02 0.47   160.75   161.33   161.97   894 1.00
## mu[134]    161.20    0.01 0.39   160.70   161.17   161.70  1319 1.00
## mu[135]    153.02    0.01 0.36   152.57   153.01   153.46  2028 1.00
## mu[136]    160.95    0.01 0.37   160.47   160.93   161.44  1567 1.00
## mu[137]    149.03    0.01 0.37   148.56   149.03   149.49  1859 1.00
## mu[138]    156.77    0.01 0.47   156.18   156.76   157.38  1040 1.00
## mu[139]    139.15    0.02 0.53   138.47   139.14   139.84   861 1.00
## mu[140]    159.01    0.03 0.83   157.95   159.00   160.09   636 1.00
## mu[141]     87.68    0.01 0.53    87.00    87.67    88.34  2390 1.00
## mu[142]    160.58    0.01 0.36   160.11   160.57   161.06  1823 1.00
## mu[143]    149.92    0.01 0.36   149.47   149.93   150.38  1926 1.00
## mu[144]    152.94    0.01 0.47   152.34   152.93   153.53  1168 1.00
## mu[145]    118.40    0.01 0.38   117.92   118.38   118.88  1332 1.00
## mu[146]    147.04    0.01 0.37   146.56   147.05   147.51  1650 1.00
## mu[147]    152.69    0.01 0.36   152.24   152.68   153.14  2031 1.00
## mu[148]    112.55    0.01 0.39   112.06   112.54   113.04  1544 1.00
## mu[149]    108.28    0.01 0.49   107.66   108.28   108.93  1124 1.00
## mu[150]    158.87    0.01 0.38   158.38   158.85   159.36  1893 1.00
## mu[151]    156.21    0.01 0.42   155.68   156.20   156.74  1336 1.00
## mu[152]    116.27    0.01 0.38   115.79   116.25   116.76  1402 1.00
## mu[153]    114.43    0.02 0.49   113.78   114.42   115.09  1012 1.00
## mu[154]    161.01    0.02 0.56   160.30   161.01   161.76   748 1.00
## mu[155]    141.37    0.01 0.39   140.87   141.37   141.87  1323 1.00
## mu[156]    151.56    0.01 0.36   151.11   151.56   152.01  2005 1.00
## mu[157]     92.90    0.01 0.48    92.29    92.90    93.52  2375 1.00
## mu[158]    161.32    0.01 0.41   160.80   161.30   161.86  1145 1.00
## mu[159]    160.93    0.01 0.37   160.46   160.92   161.42  1580 1.00
## mu[160]    156.97    0.02 0.62   156.21   156.96   157.81   773 1.00
## mu[161]    161.23    0.02 0.51   160.59   161.23   161.92   812 1.00
## mu[162]    160.97    0.02 0.57   160.24   160.97   161.72   740 1.00
## mu[163]    152.79    0.01 0.36   152.34   152.78   153.23  2030 1.00
## mu[164]    119.33    0.01 0.38   118.85   119.31   119.81  1305 1.00
## mu[165]    151.61    0.01 0.36   151.16   151.61   152.07  2007 1.00
## mu[166]    161.09    0.01 0.38   160.61   161.07   161.59  1432 1.00
## mu[167]    154.37    0.01 0.36   153.92   154.37   154.84  1947 1.00
## mu[168]    159.69    0.01 0.37   159.21   159.68   160.17  2010 1.00
## mu[169]    152.54    0.01 0.36   152.09   152.54   152.99  2030 1.00
## mu[170]    157.00    0.02 0.53   156.34   156.99   157.70   885 1.00
## mu[171]     91.22    0.01 0.50    90.59    91.21    91.84  2393 1.00
## mu[172]     92.06    0.01 0.49    91.44    92.06    92.68  2386 1.00
## mu[173]    154.22    0.01 0.36   153.77   154.22   154.68  1963 1.00
## mu[174]    140.05    0.01 0.39   139.56   140.04   140.56  1282 1.00
## mu[175]    160.53    0.02 0.64   159.71   160.51   161.37   691 1.00
## mu[176]    157.02    0.02 0.58   156.30   157.01   157.80   810 1.00
## mu[177]    146.52    0.01 0.38   146.04   146.52   146.99  1606 1.00
## mu[178]    160.88    0.01 0.37   160.41   160.86   161.36  1626 1.00
## mu[179]    160.45    0.02 0.65   159.62   160.44   161.31   685 1.00
## mu[180]    152.66    0.01 0.36   152.21   152.66   153.11  2030 1.00
## mu[181]    161.19    0.01 0.39   160.70   161.16   161.69  1330 1.00
## mu[182]    155.65    0.01 0.39   155.16   155.63   156.15  1573 1.00
## mu[183]    157.55    0.04 0.98   156.27   157.54   158.82   620 1.00
## mu[184]    160.92    0.01 0.37   160.44   160.90   161.40  1593 1.00
## mu[185]    143.22    0.02 0.52   142.55   143.21   143.90   894 1.00
## mu[186]    140.05    0.01 0.39   139.56   140.04   140.56  1282 1.00
## mu[187]    122.75    0.01 0.38   122.27   122.74   123.23  1230 1.00
## mu[188]    128.64    0.02 0.52   127.97   128.64   129.32   866 1.00
## mu[189]    121.50    0.02 0.50   120.85   121.50   122.16   930 1.00
## mu[190]    144.88    0.01 0.38   144.39   144.88   145.37  1488 1.00
## mu[191]    159.52    0.01 0.37   159.04   159.51   160.00  1998 1.00
## mu[192]    154.06    0.01 0.36   153.61   154.06   154.52  1978 1.00
## mu[193]    155.13    0.01 0.38   154.66   155.12   155.61  1757 1.00
## mu[194]    149.61    0.01 0.36   149.15   149.62   150.07  1909 1.00
## mu[195]    151.83    0.01 0.36   151.38   151.82   152.28  2015 1.00
## mu[196]     92.44    0.01 0.49    91.82    92.43    93.06  2381 1.00
## mu[197]    154.95    0.01 0.37   154.48   154.94   155.43  1815 1.00
## mu[198]    161.29    0.02 0.49   160.67   161.29   161.96   846 1.00
## mu[199]    157.03    0.02 0.57   156.32   157.02   157.78   830 1.00
## mu[200]    112.84    0.02 0.49   112.21   112.84   113.50  1039 1.00
## mu[201]     98.82    0.01 0.51    98.14    98.83    99.48  1369 1.00
## mu[202]    149.71    0.01 0.36   149.25   149.71   150.16  1914 1.00
## mu[203]     94.80    0.01 0.53    94.10    94.80    95.48  1493 1.00
## mu[204]    159.55    0.01 0.37   159.07   159.54   160.03  2001 1.00
## mu[205]    151.08    0.01 0.36   150.63   151.08   151.54  1985 1.00
## mu[206]    161.35    0.01 0.42   160.82   161.32   161.89  1097 1.00
## mu[207]    149.23    0.01 0.36   148.76   149.23   149.68  1879 1.00
## mu[208]    154.20    0.01 0.45   153.63   154.20   154.78  1245 1.00
## mu[209]    154.37    0.01 0.36   153.92   154.37   154.84  1947 1.00
## mu[210]    154.51    0.01 0.45   153.95   154.51   155.08  1268 1.00
## mu[211]    138.08    0.01 0.39   137.59   138.07   138.59  1235 1.00
## mu[212]    155.76    0.01 0.40   155.26   155.74   156.27  1533 1.00
## mu[213]    119.69    0.01 0.38   119.21   119.67   120.17  1296 1.00
## mu[214]    155.65    0.01 0.43   155.09   155.64   156.20  1370 1.00
## mu[215]    161.19    0.01 0.39   160.70   161.16   161.69  1330 1.00
## mu[216]    151.93    0.01 0.36   151.48   151.93   152.38  2018 1.00
## mu[217]    110.63    0.01 0.49   110.01   110.63   111.29  1079 1.00
## mu[218]    161.15    0.02 0.53   160.46   161.14   161.85   779 1.00
## mu[219]    154.79    0.01 0.37   154.33   154.78   155.27  1862 1.00
## mu[220]    154.63    0.01 0.37   154.17   154.63   155.11  1907 1.00
## mu[221]    100.92    0.01 0.50   100.26   100.92   101.57  1307 1.00
## mu[222]    114.85    0.02 0.49   114.20   114.84   115.51  1006 1.00
## mu[223]    149.94    0.02 0.49   149.31   149.93   150.58  1024 1.00
## mu[224]    159.27    0.01 0.37   158.79   159.26   159.76  1966 1.00
## mu[225]    160.64    0.01 0.36   160.18   160.63   161.12  1789 1.00
## mu[226]    155.19    0.01 0.38   154.72   155.18   155.67  1736 1.00
## mu[227]    119.64    0.02 0.50   119.00   119.64   120.30   946 1.00
## mu[228]    160.75    0.01 0.37   160.29   160.73   161.23  1722 1.00
## mu[229]    151.85    0.01 0.36   151.40   151.85   152.30  2016 1.00
## mu[230]    148.93    0.01 0.37   148.47   148.94   149.39  1848 1.00
## mu[231]    152.69    0.01 0.36   152.24   152.68   153.14  2031 1.00
## mu[232]    106.54    0.01 0.40   106.04   106.53   107.03  1824 1.00
## mu[233]    148.73    0.01 0.37   148.26   148.74   149.19  1824 1.00
## mu[234]    160.72    0.01 0.37   160.26   160.70   161.20  1741 1.00
## mu[235]    149.52    0.01 0.36   149.06   149.52   149.97  1904 1.00
## mu[236]    155.05    0.01 0.44   154.49   155.04   155.61  1312 1.00
## mu[237]    156.85    0.02 0.49   156.24   156.84   157.48   997 1.00
## mu[238]    161.20    0.02 0.52   160.54   161.20   161.90   798 1.00
## mu[239]    155.15    0.01 0.38   154.68   155.14   155.63  1749 1.00
## mu[240]    151.45    0.01 0.36   150.99   151.45   151.90  2001 1.00
## mu[241]    130.38    0.01 0.39   129.89   130.37   130.88  1166 1.00
## mu[242]    150.08    0.01 0.36   149.62   150.08   150.54  1935 1.00
## mu[243]    114.47    0.01 0.38   113.98   114.45   114.96  1472 1.00
## mu[244]    157.90    0.01 0.39   157.40   157.90   158.42  1698 1.00
## mu[245]    139.47    0.01 0.39   138.97   139.46   139.98  1267 1.00
## mu[246]    119.87    0.02 0.50   119.22   119.86   120.52   944 1.00
## mu[247]    159.40    0.01 0.37   158.92   159.39   159.89  1985 1.00
## mu[248]    148.49    0.01 0.37   148.03   148.50   148.95  1797 1.00
## mu[249]    101.54    0.01 0.42   101.00   101.53   102.07  2081 1.00
## mu[250]    145.99    0.01 0.38   145.50   145.99   146.46  1564 1.00
## mu[251]    151.56    0.01 0.36   151.11   151.56   152.01  2005 1.00
## mu[252]    160.57    0.01 0.36   160.10   160.56   161.05  1829 1.00
## mu[253]    160.49    0.01 0.36   160.02   160.49   160.97  1866 1.00
## mu[254]    155.56    0.01 0.39   155.08   155.55   156.06  1604 1.00
## mu[255]    156.17    0.01 0.42   155.62   156.16   156.71  1428 1.00
## mu[256]    146.67    0.01 0.37   146.19   146.67   147.14  1618 1.00
## mu[257]    157.82    0.01 0.40   157.31   157.81   158.33  1681 1.00
## mu[258]    160.65    0.01 0.36   160.19   160.64   161.13  1783 1.00
## mu[259]    146.82    0.01 0.37   146.34   146.82   147.29  1631 1.00
## mu[260]    160.60    0.02 0.63   159.80   160.59   161.43   696 1.00
## mu[261]    149.26    0.01 0.36   148.80   149.26   149.71  1883 1.00
## mu[262]    160.40    0.01 0.36   159.94   160.40   160.88  1904 1.00
## mu[263]    146.14    0.01 0.38   145.65   146.14   146.61  1575 1.00
## mu[264]     96.22    0.01 0.46    95.64    96.21    96.79  2300 1.00
## mu[265]    151.93    0.01 0.36   151.48   151.93   152.38  2018 1.00
## mu[266]     99.17    0.01 0.44    98.62    99.17    99.72  2190 1.00
## mu[267]    154.65    0.01 0.37   154.19   154.65   155.12  1902 1.00
## mu[268]    161.23    0.01 0.39   160.72   161.21   161.74  1280 1.00
## mu[269]    160.99    0.02 0.57   160.27   160.99   161.74   744 1.00
## mu[270]    148.36    0.01 0.37   147.89   148.36   148.82  1782 1.00
## mu[271]    149.50    0.02 0.50   148.87   149.49   150.14  1011 1.00
## mu[272]    159.76    0.03 0.74   158.81   159.74   160.73   654 1.00
## mu[273]    117.09    0.01 0.38   116.60   117.06   117.57  1374 1.00
## mu[274]    154.43    0.01 0.36   153.97   154.42   154.90  1941 1.00
## mu[275]    158.76    0.01 0.38   158.27   158.75   159.26  1871 1.00
## mu[276]    154.52    0.01 0.37   154.07   154.52   154.99  1929 1.00
## mu[277]    158.90    0.01 0.38   158.41   158.89   159.40  1900 1.00
## mu[278]    151.17    0.01 0.36   150.71   151.17   151.63  1989 1.00
## mu[279]    154.20    0.01 0.36   153.75   154.20   154.66  1965 1.00
## mu[280]    124.30    0.01 0.39   123.82   124.29   124.78  1206 1.00
## mu[281]    143.59    0.01 0.38   143.10   143.59   144.08  1416 1.00
## mu[282]    141.00    0.01 0.39   140.51   140.99   141.50  1311 1.00
## mu[283]    145.79    0.01 0.38   145.30   145.79   146.27  1549 1.00
## mu[284]    161.06    0.01 0.38   160.58   161.04   161.56  1463 1.00
## mu[285]    161.34    0.02 0.47   160.74   161.33   161.97   886 1.00
## mu[286]    151.64    0.01 0.48   151.03   151.63   152.26  1107 1.00
## mu[287]    151.20    0.01 0.36   150.74   151.19   151.66  1990 1.00
## mu[288]    161.25    0.01 0.40   160.74   161.23   161.77  1248 1.00
## mu[289]    158.46    0.01 0.39   157.97   158.46   158.97  1811 1.00
## mu[290]    142.63    0.01 0.39   142.14   142.63   143.13  1372 1.00
## mu[291]    116.79    0.01 0.38   116.31   116.77   117.28  1384 1.00
## mu[292]    103.60    0.01 0.41   103.07   103.59   104.11  1977 1.00
## mu[293]    161.13    0.01 0.38   160.64   161.11   161.63  1390 1.00
## mu[294]    142.32    0.01 0.39   141.83   142.32   142.82  1359 1.00
## mu[295]    158.38    0.01 0.39   157.89   158.38   158.89  1794 1.00
## mu[296]    156.31    0.01 0.42   155.77   156.29   156.85  1283 1.00
## mu[297]    153.21    0.01 0.36   152.76   153.20   153.66  2025 1.00
## mu[298]    148.56    0.02 0.50   147.91   148.55   149.21   985 1.00
## mu[299]    123.57    0.01 0.39   123.09   123.55   124.04  1217 1.00
## mu[300]    142.81    0.01 0.39   142.32   142.81   143.31  1379 1.00
## mu[301]     88.93    0.01 0.52    88.27    88.93    89.58  2397 1.00
## mu[302]    151.93    0.01 0.36   151.48   151.93   152.38  2018 1.00
## mu[303]    154.72    0.01 0.44   154.16   154.72   155.29  1285 1.00
## mu[304]    145.79    0.01 0.38   145.30   145.79   146.27  1549 1.00
## mu[305]    161.29    0.01 0.40   160.78   161.27   161.83  1193 1.00
## mu[306]    155.24    0.01 0.38   154.77   155.23   155.72  1720 1.00
## mu[307]    126.27    0.01 0.39   125.79   126.25   126.76  1184 1.00
## mu[308]     99.53    0.01 0.44    98.98    99.52   100.07  2175 1.00
## mu[309]    156.64    0.01 0.45   156.06   156.62   157.22  1113 1.00
## mu[310]    159.57    0.03 0.77   158.59   159.55   160.57   648 1.00
## mu[311]    153.75    0.01 0.36   153.31   153.75   154.21  2001 1.00
## mu[312]    111.37    0.01 0.49   110.74   111.37   112.03  1067 1.00
## mu[313]    161.37    0.01 0.43   160.83   161.35   161.94  1025 1.00
## mu[314]    161.27    0.01 0.40   160.75   161.24   161.79  1232 1.00
## mu[315]    143.33    0.01 0.39   142.84   143.33   143.83  1403 1.00
## mu[316]    150.02    0.01 0.36   149.56   150.02   150.48  1931 1.00
## mu[317]    159.14    0.03 0.82   158.09   159.12   160.20   638 1.00
## mu[318]    155.34    0.01 0.44   154.78   155.33   155.89  1339 1.00
## mu[319]    161.37    0.01 0.44   160.82   161.36   161.95   999 1.00
## mu[320]    150.65    0.01 0.36   150.19   150.65   151.11  1964 1.00
## mu[321]    158.44    0.01 0.39   157.95   158.44   158.95  1807 1.00
## mu[322]    156.80    0.03 0.68   155.96   156.79   157.71   731 1.00
## mu[323]    154.84    0.01 0.37   154.38   154.84   155.32  1846 1.00
## mu[324]     91.14    0.01 0.55    90.44    91.13    91.85  1603 1.00
## mu[325]     92.21    0.01 0.54    91.51    92.21    92.91  1572 1.00
## mu[326]     66.38    0.02 0.81    65.37    66.38    67.42  1934 1.00
## mu[327]     93.37    0.01 0.53    92.67    93.37    94.07  1537 1.00
## mu[328]    108.42    0.01 0.39   107.93   108.41   108.91  1729 1.00
## mu[329]     86.51    0.01 0.55    85.82    86.50    87.20  2379 1.00
## mu[330]    151.98    0.01 0.36   151.53   151.98   152.43  2019 1.00
## mu[331]    102.32    0.01 0.42   101.78   102.31   102.84  2043 1.00
## mu[332]     69.89    0.02 0.76    68.94    69.89    70.87  2012 1.00
## mu[333]     92.60    0.01 0.54    91.89    92.60    93.30  1560 1.00
## mu[334]     98.37    0.01 0.44    97.81    98.37    98.93  2223 1.00
## mu[335]     85.57    0.01 0.59    84.82    85.58    86.33  1742 1.00
## mu[336]    152.49    0.01 0.36   152.04   152.49   152.94  2029 1.00
## mu[337]    157.36    0.01 0.40   156.85   157.35   157.88  1600 1.00
## mu[338]    138.26    0.02 0.53   137.58   138.25   138.95   856 1.00
## mu[339]     91.31    0.01 0.50    90.68    91.30    91.94  2392 1.00
## mu[340]    157.86    0.01 0.40   157.36   157.85   158.37  1689 1.00
## mu[341]    151.50    0.01 0.36   151.05   151.50   151.96  2003 1.00
## mu[342]    157.64    0.01 0.40   157.13   157.63   158.15  1648 1.00
## mu[343]    155.48    0.01 0.39   155.00   155.47   155.97  1635 1.00
## mu[344]    114.70    0.01 0.38   114.21   114.68   115.19  1465 1.00
## mu[345]    101.19    0.01 0.43   100.65   101.19   101.72  2098 1.00
## mu[346]    160.55    0.01 0.36   160.08   160.54   161.03  1840 1.00
## mu[347]    153.02    0.01 0.36   152.57   153.01   153.46  2028 1.00
## mu[348]    104.79    0.01 0.41   104.27   104.77   105.28  1915 1.00
## mu[349]     96.97    0.01 0.52    96.28    96.97    97.64  1426 1.00
## mu[350]    160.77    0.01 0.37   160.31   160.76   161.26  1703 1.00
## mu[351]    159.64    0.01 0.37   159.16   159.64   160.12  2007 1.00
## mu[352]    161.36    0.02 0.46   160.77   161.35   161.97   914 1.00
## mu[353]    160.44    0.01 0.36   159.97   160.43   160.92  1890 1.00
## mu[354]     91.78    0.01 0.49    91.16    91.77    92.40  2388 1.00
## mu[355]     69.84    0.02 0.76    68.88    69.83    70.80  1839 1.00
## mu[356]    138.38    0.02 0.53   137.70   138.36   139.06   856 1.00
## mu[357]    153.82    0.01 0.36   153.37   153.81   154.28  1997 1.00
## mu[358]    153.73    0.01 0.36   153.28   153.73   154.19  2002 1.00
## mu[359]     91.34    0.01 0.55    90.63    91.33    92.04  1597 1.00
## mu[360]     87.68    0.01 0.53    87.00    87.67    88.34  2390 1.00
## mu[361]     77.04    0.02 0.68    76.18    77.05    77.90  1847 1.00
## mu[362]     91.03    0.01 0.50    90.40    91.02    91.66  2394 1.00
## mu[363]     71.41    0.02 0.74    70.49    71.41    72.35  2048 1.00
## mu[364]     94.23    0.01 0.53    93.53    94.23    94.92  1511 1.00
## mu[365]     73.65    0.02 0.70    72.77    73.65    74.56  2103 1.00
## mu[366]    156.99    0.02 0.61   156.24   156.98   157.81   783 1.00
## mu[367]    154.60    0.01 0.37   154.14   154.59   155.07  1916 1.00
## mu[368]     94.90    0.01 0.53    94.20    94.90    95.58  1490 1.00
## mu[369]     95.37    0.01 0.52    94.68    95.37    96.04  1475 1.00
## mu[370]     72.05    0.02 0.73    71.15    72.05    72.99  2064 1.00
## mu[371]     78.88    0.01 0.64    78.08    78.89    79.70  2232 1.00
## mu[372]    160.29    0.01 0.36   159.82   160.28   160.76  1944 1.00
## mu[373]    133.47    0.01 0.39   132.97   133.46   133.97  1176 1.00
## mu[374]    139.13    0.01 0.39   138.63   139.12   139.64  1258 1.00
## mu[375]    160.64    0.02 0.63   159.84   160.63   161.47   700 1.00
## mu[376]    117.51    0.02 0.49   116.87   117.51   118.17   969 1.00
## mu[377]     80.63    0.01 0.64    79.82    80.64    81.44  1820 1.00
## mu[378]     75.56    0.01 0.68    74.70    75.55    76.43  2150 1.00
## mu[379]     87.38    0.01 0.54    86.71    87.38    88.06  2388 1.00
## mu[380]    160.85    0.02 0.59   160.09   160.85   161.64   723 1.00
## mu[381]     74.29    0.02 0.70    73.41    74.29    75.19  2118 1.00
## mu[382]    130.56    0.01 0.39   130.07   130.55   131.06  1166 1.00
## mu[383]    160.40    0.01 0.36   159.94   160.40   160.88  1904 1.00
## mu[384]     88.88    0.01 0.56    88.16    88.88    89.60  1665 1.00
## mu[385]     95.30    0.01 0.47    94.72    95.30    95.89  2326 1.00
## mu[386]    154.90    0.01 0.37   154.43   154.89   155.37  1831 1.00
## mu[387]    155.11    0.01 0.38   154.64   155.10   155.59  1763 1.00
## mu[388]    156.54    0.01 0.44   155.98   156.53   157.11  1162 1.00
## mu[389]    160.00    0.01 0.36   159.52   159.99   160.49  2001 1.00
## mu[390]    113.48    0.01 0.39   112.99   113.46   113.97  1508 1.00
## mu[391]    112.28    0.02 0.49   111.64   112.27   112.93  1050 1.00
## mu[392]     89.28    0.01 0.56    88.56    89.27    90.00  1655 1.00
## mu[393]    154.06    0.01 0.36   153.61   154.06   154.52  1978 1.00
## mu[394]    134.78    0.02 0.53   134.10   134.77   135.46   850 1.00
## mu[395]    114.85    0.02 0.49   114.20   114.84   115.51  1006 1.00
## mu[396]     94.75    0.01 0.47    94.17    94.74    95.35  2339 1.00
## mu[397]    160.77    0.02 0.60   160.00   160.76   161.57   713 1.00
## mu[398]    150.77    0.01 0.36   150.31   150.77   151.23  1970 1.00
## mu[399]    133.51    0.02 0.53   132.83   133.50   134.18   851 1.00
## mu[400]    156.94    0.02 0.51   156.31   156.93   157.59   941 1.00
## mu[401]     83.23    0.01 0.61    82.44    83.23    84.00  1785 1.00
## mu[402]     77.14    0.02 0.67    76.29    77.15    78.01  1846 1.00
## mu[403]    159.87    0.01 0.37   159.38   159.86   160.35  2010 1.00
## mu[404]    111.53    0.01 0.39   111.05   111.52   112.02  1586 1.00
## mu[405]    152.67    0.01 0.47   152.07   152.65   153.26  1155 1.00
## mu[406]    161.01    0.01 0.38   160.53   161.00   161.51  1508 1.00
## mu[407]     71.84    0.02 0.73    70.93    71.84    72.78  2058 1.00
## mu[408]    136.84    0.02 0.53   136.16   136.84   137.52   852 1.00
## mu[409]     78.10    0.02 0.66    77.26    78.11    78.96  1841 1.00
## mu[410]    154.22    0.01 0.36   153.77   154.22   154.68  1963 1.00
## mu[411]    161.31    0.01 0.41   160.79   161.28   161.84  1174 1.00
## mu[412]    155.79    0.01 0.43   155.24   155.78   156.34  1384 1.00
## mu[413]    147.91    0.01 0.37   147.43   147.91   148.37  1734 1.00
## mu[414]    161.35    0.01 0.42   160.82   161.33   161.91  1079 1.00
## mu[415]    154.69    0.01 0.37   154.23   154.68   155.16  1892 1.00
## mu[416]     74.50    0.02 0.69    73.63    74.50    75.39  2124 1.00
## mu[417]    153.71    0.01 0.36   153.26   153.71   154.17  2004 1.00
## mu[418]    160.07    0.01 0.36   159.59   160.06   160.55  1992 1.00
## mu[419]    150.20    0.01 0.36   149.74   150.21   150.66  1941 1.00
## mu[420]    144.14    0.01 0.38   143.65   144.14   144.63  1444 1.00
## mu[421]    124.74    0.02 0.51   124.07   124.73   125.40   895 1.00
## mu[422]    154.27    0.01 0.45   153.70   154.26   154.84  1250 1.00
## mu[423]    147.77    0.01 0.37   147.29   147.77   148.23  1719 1.00
## mu[424]    160.83    0.01 0.37   160.37   160.81   161.31  1665 1.00
## mu[425]    145.32    0.01 0.38   144.83   145.32   145.80  1516 1.00
## mu[426]     83.84    0.01 0.61    83.06    83.85    84.62  1775 1.00
## mu[427]    158.28    0.01 0.39   157.78   158.28   158.78  1773 1.00
## mu[428]    160.67    0.01 0.37   160.21   160.66   161.15  1771 1.00
## mu[429]    140.18    0.02 0.53   139.50   140.17   140.87   868 1.00
## mu[430]    161.30    0.01 0.41   160.79   161.28   161.83  1183 1.00
## mu[431]    154.14    0.01 0.36   153.69   154.14   154.60  1971 1.00
## mu[432]    137.13    0.02 0.53   136.45   137.12   137.81   853 1.00
## mu[433]    118.43    0.02 0.50   117.79   118.43   119.09   959 1.00
## mu[434]    160.67    0.01 0.37   160.21   160.66   161.15  1771 1.00
## mu[435]    154.14    0.01 0.45   153.57   154.14   154.72  1241 1.00
## mu[436]    160.06    0.01 0.36   159.58   160.05   160.54  1994 1.00
## mu[437]    160.65    0.02 0.62   159.85   160.64   161.47   701 1.00
## mu[438]    160.80    0.01 0.37   160.34   160.79   161.28  1684 1.00
## mu[439]    156.34    0.01 0.43   155.80   156.32   156.88  1267 1.00
## mu[440]     97.34    0.01 0.51    96.65    97.35    98.02  1414 1.00
## mu[441]     69.35    0.02 0.76    68.39    69.35    70.34  2000 1.00
## mu[442]    160.40    0.01 0.36   159.94   160.40   160.88  1904 1.00
## mu[443]    145.24    0.01 0.38   144.75   145.24   145.72  1511 1.00
## mu[444]    146.67    0.01 0.37   146.19   146.67   147.15  1618 1.00
## mu[445]    156.57    0.01 0.45   156.00   156.55   157.14  1149 1.00
## mu[446]    161.33    0.01 0.41   160.81   161.30   161.87  1135 1.00
## mu[447]    159.98    0.01 0.36   159.50   159.97   160.46  2003 1.00
## mu[448]    147.15    0.01 0.37   146.67   147.16   147.62  1660 1.00
## mu[449]    130.29    0.02 0.52   129.62   130.28   130.95   858 1.00
## mu[450]    106.75    0.01 0.49   106.11   106.74   107.39  1157 1.00
## mu[451]    160.82    0.01 0.37   160.36   160.80   161.30  1671 1.00
## mu[452]    148.49    0.01 0.37   148.03   148.50   148.95  1797 1.00
## mu[453]     76.29    0.01 0.67    75.45    76.29    77.15  2168 1.00
## mu[454]    160.32    0.01 0.36   159.85   160.32   160.80  1933 1.00
## mu[455]    132.90    0.01 0.39   132.40   132.89   133.40  1173 1.00
## mu[456]     74.93    0.01 0.69    74.06    74.92    75.81  2134 1.00
## mu[457]     75.75    0.02 0.69    74.87    75.76    76.63  1851 1.00
## mu[458]    160.96    0.02 0.57   160.23   160.96   161.72   739 1.00
## mu[459]    144.87    0.02 0.52   144.21   144.85   145.53   915 1.00
## mu[460]    161.22    0.02 0.51   160.56   161.21   161.91   804 1.00
## mu[461]    160.76    0.01 0.37   160.30   160.75   161.25  1709 1.00
## mu[462]    153.45    0.01 0.36   153.01   153.45   153.91  2017 1.00
## mu[463]     71.62    0.02 0.73    70.71    71.62    72.56  2053 1.00
## mu[464]    143.24    0.01 0.39   142.75   143.25   143.74  1399 1.00
## mu[465]    156.80    0.01 0.48   156.20   156.78   157.41  1024 1.00
## mu[466]    156.30    0.01 0.42   155.75   156.29   156.84  1444 1.00
## mu[467]     62.24    0.02 0.87    61.14    62.23    63.35  1850 1.00
## mu[468]     72.05    0.02 0.73    71.11    72.05    72.98  1849 1.00
## mu[469]    154.37    0.01 0.36   153.92   154.37   154.84  1947 1.00
## mu[470]    142.10    0.01 0.39   141.61   142.10   142.59  1350 1.00
## mu[471]    104.28    0.01 0.41   103.76   104.27   104.78  1942 1.00
## mu[472]     75.24    0.01 0.68    74.38    75.24    76.12  2142 1.00
## mu[473]    160.32    0.01 0.36   159.85   160.32   160.80  1933 1.00
## mu[474]    155.43    0.01 0.38   154.96   155.42   155.93  1651 1.00
## mu[475]    113.63    0.01 0.38   113.14   113.61   114.12  1503 1.00
## mu[476]     95.56    0.01 0.52    94.87    95.56    96.23  1469 1.00
## mu[477]    160.37    0.01 0.36   159.90   160.36   160.85  1917 1.00
## mu[478]    151.53    0.01 0.36   151.08   151.53   151.99  2004 1.00
## mu[479]    161.22    0.01 0.39   160.71   161.19   161.72  1296 1.00
## mu[480]    154.88    0.01 0.37   154.42   154.87   155.36  1836 1.00
## mu[481]     87.38    0.01 0.54    86.71    87.38    88.06  2388 1.00
## mu[482]    157.13    0.01 0.41   156.60   157.12   157.65  1561 1.00
## mu[483]    152.27    0.01 0.36   151.82   152.26   152.71  2026 1.00
## mu[484]     83.02    0.01 0.61    82.23    83.03    83.79  1788 1.00
## mu[485]    159.27    0.01 0.37   158.79   159.26   159.76  1966 1.00
## mu[486]    156.99    0.02 0.52   156.33   156.97   157.67   902 1.00
## mu[487]    120.69    0.02 0.50   120.04   120.69   121.34   937 1.00
## mu[488]     91.63    0.01 0.54    90.92    91.62    92.33  1589 1.00
## mu[489]    160.99    0.01 0.37   160.51   160.98   161.48  1528 1.00
## mu[490]    132.10    0.01 0.39   131.60   132.08   132.59  1169 1.00
## mu[491]    161.32    0.01 0.41   160.80   161.30   161.86  1145 1.00
## mu[492]    154.04    0.01 0.36   153.59   154.04   154.50  1980 1.00
## mu[493]    110.96    0.01 0.49   110.33   110.96   111.62  1074 1.00
## mu[494]    145.20    0.01 0.38   144.71   145.20   145.68  1508 1.00
## mu[495]    143.33    0.01 0.39   142.84   143.33   143.83  1403 1.00
## mu[496]    153.31    0.01 0.36   152.87   153.31   153.77  2022 1.00
## mu[497]    140.62    0.01 0.39   140.13   140.62   141.13  1299 1.00
## mu[498]    160.10    0.01 0.36   159.62   160.09   160.58  1988 1.00
## mu[499]    156.81    0.01 0.41   156.28   156.80   157.34  1513 1.00
## mu[500]    153.02    0.01 0.47   152.42   153.01   153.61  1173 1.00
## mu[501]    157.98    0.01 0.39   157.47   157.97   158.49  1712 1.00
## mu[502]    161.29    0.02 0.49   160.67   161.29   161.96   846 1.00
## mu[503]    153.39    0.01 0.36   152.94   153.38   153.84  2019 1.00
## mu[504]    121.57    0.02 0.50   120.92   121.57   122.23   930 1.00
## mu[505]     86.70    0.01 0.54    86.02    86.70    87.39  2381 1.00
## mu[506]    154.78    0.01 0.44   154.22   154.78   155.35  1290 1.00
## mu[507]    153.67    0.01 0.36   153.22   153.67   154.13  2006 1.00
## mu[508]     99.97    0.01 0.43    99.42    99.96   100.51  2155 1.00
## mu[509]    122.52    0.02 0.50   121.87   122.52   123.18   920 1.00
## mu[510]    161.37    0.01 0.45   160.79   161.36   161.97   939 1.00
## mu[511]    148.96    0.01 0.37   148.50   148.97   149.42  1852 1.00
## mu[512]    111.14    0.01 0.39   110.65   111.13   111.63  1603 1.00
## mu[513]     89.97    0.01 0.56    89.25    89.96    90.68  1636 1.00
## mu[514]     93.65    0.01 0.48    93.05    93.64    94.25  2363 1.00
## mu[515]    153.25    0.01 0.36   152.80   153.25   153.70  2023 1.00
## mu[516]    145.71    0.01 0.38   145.23   145.72   146.20  1544 1.00
## mu[517]    154.30    0.01 0.36   153.84   154.29   154.76  1956 1.00
## mu[518]     59.84    0.02 0.91    58.70    59.83    61.00  1806 1.00
## mu[519]    148.53    0.01 0.37   148.06   148.53   148.98  1801 1.00
## mu[520]     63.48    0.02 0.85    62.40    63.47    64.57  1874 1.00
## mu[521]    100.65    0.01 0.50    99.98   100.66   101.31  1314 1.00
## mu[522]    125.79    0.02 0.51   125.12   125.78   126.45   886 1.00
## mu[523]     84.04    0.01 0.57    83.31    84.04    84.77  2343 1.00
## mu[524]    158.56    0.01 0.38   158.07   158.55   159.06  1831 1.00
## mu[525]    154.16    0.01 0.36   153.71   154.16   154.62  1969 1.00
## mu[526]    129.96    0.02 0.52   129.29   129.96   130.63   860 1.00
## mu[527]    119.90    0.01 0.38   119.42   119.88   120.39  1291 1.00
## mu[528]     96.40    0.01 0.46    95.82    96.39    96.97  2294 1.00
## mu[529]    160.56    0.01 0.36   160.09   160.55   161.04  1834 1.00
## mu[530]    150.05    0.01 0.36   149.59   150.05   150.51  1933 1.00
## mu[531]    125.16    0.01 0.39   124.68   125.14   125.65  1196 1.00
## mu[532]     84.04    0.01 0.57    83.31    84.04    84.77  2343 1.00
## mu[533]    156.06    0.01 0.43   155.51   156.05   156.60  1415 1.00
## mu[534]    151.50    0.01 0.36   151.05   151.50   151.96  2003 1.00
## mu[535]    155.87    0.01 0.40   155.37   155.86   156.39  1489 1.00
## mu[536]    110.46    0.01 0.49   109.84   110.46   111.12  1082 1.00
## mu[537]     71.52    0.02 0.73    70.60    71.51    72.46  2051 1.00
## mu[538]    143.33    0.01 0.39   142.84   143.33   143.83  1403 1.00
## mu[539]     78.84    0.02 0.66    78.02    78.85    79.69  1836 1.00
## mu[540]    142.52    0.02 0.53   141.85   142.51   143.20   887 1.00
## mu[541]    161.37    0.01 0.44   160.82   161.36   161.95   999 1.00
## mu[542]    156.90    0.02 0.65   156.09   156.89   157.77   749 1.00
## mu[543]     78.95    0.02 0.65    78.12    78.96    79.79  1835 1.00
## mu[544]    161.37    0.01 0.45   160.80   161.36   161.96   955 1.00
## lp__     -1189.58    0.06 1.61 -1191.77 -1189.22 -1187.88   694 1.01
## 
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 15:25:58 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).